Apuntes de muestreo
M Sc. Mario Gregorio Saavedra Rodríguez
17/9/2019
1 Libreria TeachingSamplig
- Con este código cargo la librería
library(TeachingSampling)2 Marco de datos
data(Lucy)Sin subcobertura
Sin sobrecobertura
Sin duplicados
2.1 Subconjunto del marco muestral
Lucy[1:6, 1:2]## ID Ubication
## 1 AB001 c1k1
## 2 AB002 c1k2
## 3 AB003 c1k3
## 4 AB004 c1k4
## 5 AB005 c1k5
## 6 AB006 c1k6Lucy[2390:2396, 3:5]## Level Zone Income
## 2390 Big E 1220
## 2391 Big E 1030
## 2392 Big E 1020
## 2393 Big E 1077
## 2394 Big E 1297
## 2395 Big E 1640
## 2396 Big E 18602.2 Variables en el marco de muestreo
names(Lucy)## [1] "ID" "Ubication" "Level" "Zone" "Income" "Employees"
## [7] "Taxes" "SPAM"Identificación: ID
Ubicación: Ubication
Variables e información auxiliar: Income, Employes, Taxes, SPAM, Zone
colnames(Lucy)## [1] "ID" "Ubication" "Level" "Zone" "Income" "Employees"
## [7] "Taxes" "SPAM"2.3 Tamaño poblacional
nrow(Lucy)## [1] 23962.4 Información disponible
ncol(Lucy)## [1] 82.5 Dimensiones del marco de muestreo
dim(Lucy)## [1] 2396 82.6 Estadísticas descriptivas básicas
2.6.1 Income: ingreso
2.6.1.1 Medidas de localización
cat("La media del ingreso es:",mean(Lucy$Income))## La media del ingreso es: 432.06052.6.1.2 Medidas de dispersión
cat("La desviación estándar del ingreso es:",sd(Lucy$Income))## La desviación estándar del ingreso es: 266.97922.6.2 Employes: empleados
2.6.2.1 Medidas de localización
cat("La media del número de empleados es:",mean(Lucy$Employees))## La media del número de empleados es: 63.41822.6.2.2 Medidas de dispersión
cat("El desviación estándar del número de empleados es:",sd(Lucy$Employees))## El desviación estándar del número de empleados es: 32.889942.6.3 Taxes: impuestos
2.6.3.1 Medidas de localización
cat("La media de los impuestos es:",mean(Lucy$Taxes))## La media de los impuestos es: 11.958892.6.3.2 Medidas de dispersión
cat("El desviación estándar de los impuestos es:",sd(Lucy$Taxes))## El desviación estándar de los impuestos es: 17.335082.7 Dsitribución de los datos
2.7.1 Gráfico de caja y bigotes
2.7.1.1 Income: ingreso
boxplot(Lucy$Income, horizontal=TRUE, main="Gráfico de caja y bigotes de los ingresos", col="blue")
boxplot(Lucy$Income ~ Lucy$Level, horizontal=FALSE, main="Histograma de los ingresos según tamaño", xlab="Tamaño", ylab="Ingresos", col=c("blue","red","green"))
2.7.1.2 Employees: empleados
boxplot(Lucy$Employees, horizontal=TRUE, main="Gráfico de caja y bigotes del número de empleados", col="blue")
boxplot(Lucy$Employees ~ Lucy$Level, horizontal=FALSE, main="Histograma del número de emplados según tamaño", xlab="Tamaño", ylab="Ingresos", col=c("blue","red","green"))
2.7.1.3 Taxes: impuestos
boxplot(Lucy$Taxes, horizontal=TRUE, main="Gráfico de caja y bigotes de los impuestos", col="green")
boxplot(Lucy$Taxes ~ Lucy$Level, horizontal=FALSE, main="Gráfico de caja y bigotes de los impuestos según tamaño", xlab="Tamaño", ylab="Ingresos", col=c("blue","red","green"))
2.7.2 Gráfico de barras
2.7.2.1 Level: tamaño
barplot(table(Lucy$Level), main="Gráfico de barras del tamaño de la empresa", col=c("yellow", "orange", "red"), las=1)
2.7.2.2 Zone: zona
barplot(table(Lucy$Zone), main="Gráfico de barras de la zona donde se ubica la empresa", col=c("yellow", "orange", "red", "purple", "blue"), las=1)
2.7.2.3 SPAM: correo masivo
barplot(table(Lucy$SPAM), main="Gráfico de barras de SPAM", col=c("yellow", "orange"), las=1)
2.8 Estadísticas descriptivas robustas
attach(Lucy)2.8.1 Income: ingreso
2.8.1.1 Medidas de localización
cat("La mediana de los ingresos es:",median(Income))## La mediana de los ingresos es: 3902.8.1.2 Medidas de dispersión
cat("El rango intercuartílico de los ingresos es:",IQR(Income))## El rango intercuartílico de los ingresos es: 3462.8.2 Employes: empleados
2.8.2.1 Medidas de localización
cat("La mediana del número de empleados es:",median(Employees))## La mediana del número de empleados es: 632.8.2.2 Medidas de dispersión
cat("El rango intercuartílico del número de empleados es:",IQR(Employees))## El rango intercuartílico del número de empleados es: 462.8.3 Taxes: impuestos
2.8.3.1 Medidas de localización
cat("La mediana de los impuestos es:",median(Taxes))## La mediana de los impuestos es: 72.8.3.2 Medidas de dispersión
cat("El rango intercuartílico de los impuestos es:",IQR(Taxes))## El rango intercuartílico de los impuestos es: 132.9 Dsitribución de los datos
2.9.1 Histograma
2.9.1.1 Income: ingreso
hist(Income, main="Histograma de los ingresos", xlab="Ingresos", ylab="frecuencia", col=rainbow(7))
2.9.1.2 Employees: empleados
hist(Employees, main="Histograma del número de empleados", xlab="Ingresos", ylab="frecuencia", col=rainbow(14))
2.9.1.3 Taxes: impuestos
hist(Taxes, main="Histograma de los impuestos", xlab="Ingresos", ylab="frecuencia", col=rainbow(7))
2.9.2 Gráfico de sectores
2.9.2.1 Level: tamaño
pie(table(Level), main="Graico de sectores del tamaño de la empresa", col=c("yellow", "orange", "red"), las=1, edges=2)
2.9.2.2 Zone: zona
pie(table(Zone), main="Gráfico de sectores de la zona donde se ubica la empresa", col=c("yellow", "orange", "red", "purple", "blue"), las=1, edges=23)
2.9.2.3 SPAM: correo masivo
pie(table(SPAM), main="Gráfico de sectores de SPAM", col=c("yellow", "orange"), las=1, edges=235)
#detach(Lucy)2.10 Estructura del marco muestral
str(Lucy)## 'data.frame': 2396 obs. of 8 variables:
## $ ID : Factor w/ 2396 levels "AB001","AB002",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ Ubication: Factor w/ 2396 levels "c10k1","c10k10",..: 991 1002 1013 1024 1035 1046 1057 1068 1079 992 ...
## $ Level : Factor w/ 3 levels "Big","Medium",..: 3 3 3 3 3 3 3 3 3 3 ...
## $ Zone : Factor w/ 5 levels "A","B","C","D",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Income : int 281 329 405 360 391 296 490 473 350 361 ...
## $ Employees: int 41 19 68 89 91 89 22 57 84 25 ...
## $ Taxes : num 3 4 7 5 7 3 10.5 10 5 5 ...
## $ SPAM : Factor w/ 2 levels "no","yes": 1 2 1 1 2 1 2 2 2 1 ...2.11 Resumen del marco de muestreo
summary(Lucy[,3:8],digits=1)## Level Zone Income Employees Taxes SPAM
## Big : 83 A:307 Min. : 1 Min. : 1 Min. : 0.5 no : 937
## Medium: 737 B:727 1st Qu.: 230 1st Qu.: 38 1st Qu.: 2.0 yes:1459
## Small :1576 C:974 Median : 390 Median : 63 Median : 7.0
## D:223 Mean : 432 Mean : 63 Mean : 12.0
## E:165 3rd Qu.: 576 3rd Qu.: 84 3rd Qu.: 15.0
## Max. :2510 Max. :263 Max. :305.0nrow(Lucy)## [1] 23962.12 Total y total estimado
\[\widehat{t}_{y}(S=s)=\frac{N}{n}{\sum}_{i=1}^{n}y_i=N\frac{{\sum}_{i=1}^{n}y_i}{n}=N\bar{y}{\implies}\widehat{t}_{y}(S=U)=\frac{N}{N}{\sum}_{i=1}^{N}y_i=N\frac{{\sum}_{i=1}^{N}y_i}{N}=N\mu_{y}\]
2.12.1 Función del total
total <- function(x){
nrow(Lucy)*mean(x)
}total(Income)## [1] 1035217total(Income) == sum(Income)## [1] TRUEtotal(Employees)## [1] 151950total(Employees) == sum(Employees)## [1] TRUEtotal(Taxes)## [1] 28653.5total(Taxes) == sum(Taxes)## [1] TRUE2.13 Matriz de correlaciones
cor(Lucy[,c("Income","Employees","Taxes")])## Income Employees Taxes
## Income 1.0000000 0.645536 0.9169541
## Employees 0.6455360 1.000000 0.6468550
## Taxes 0.9169541 0.646855 1.00000002.14 Gráfico de correlaciones
panel.hist <- function(x, ...)
{
usr <- par("usr"); on.exit(par(usr))
par(usr=c(usr[1:2], 0, 1.5) )
h <- hist(x, plot=FALSE)
breaks <- h$breaks; nB <- length(breaks)
y <- h$counts; y <- y/max(y)
rect(breaks[-nB], 0, breaks[-1], y, col="#adff2f", ...)
}pairs(Lucy[,c("Income","Employees","Taxes")], bg="blue", horOdd=TRUE, diag.panel=panel.hist)
panel.cor <- function(x, y, digits=2, prefix="", cex.cor, ...)
{
usr <- par("usr"); on.exit(par(usr))
par(usr=c(0, 1, 0, 1))
r <- abs(cor(x, y))
txt <- format(c(r, 0.123456789), digits=digits)[1]
txt <- paste0(prefix, txt)
if(missing(cex.cor)) cex.cor <- 0.8/strwidth(txt)
text(0.5, 0.5, txt, cex=cex.cor * r)
}pairs(Lucy[,c("Income","Employees","Taxes")], upper.panel=panel.cor, gap=0, row1attop=FALSE)
3 Población y muestra aleatoria
3.1 Población finita
\[U=\{e_1,e_2,\ldots,e_k,\ldots,e_N\}=\{1,2,\ldots,k\ldots,N\}\text{; }N<\infty\]
\[U=\{Santiago, Nestor, Nayibe, Raul, Jhon\}\]
U <- c("Santiago", "Nestor", "Nayibe", "Raul", "Jhon")4 Muestra aleatoria
\[s=\{x{\mid}x{\in}U\text{ & }0{\leq}\mathcal{P}(x{\in}s){\leq}1\text{ y conocida}\}=\{1,2,\ldots,k\ldots,n(S)\}{\subseteq}U\]
\[S:U{\rightarrow}\{1,2,\ldots,k\ldots,n(S)\}\]
4.1 Muestra aleatoria sin reemplazo
\[\boldsymbol{s}=\left(s_2,s_2,\ldots,s_N\right)^t{\in}\{0,1\}^N\]
\[ s_k= \begin{cases} 1&\text{ si el k-ésimo elemento pertenece a la muestra}\\ 0&\text{ en cualquier otro caso} \end{cases} \]
\[n(S)={\sum}_{k{\in}s}1\]
n <- 0; s <- sample(U,size=n,replace=FALSE); s## character(0)\[s=\{\}\]
n <- 1; s <- sample(U,size=n,replace=FALSE); s## [1] "Nayibe"\[s=\{Nayibe\}\]
n <- 2; s <- sample(U,size=n,replace=FALSE); s## [1] "Santiago" "Nestor"\[s=\{Santiago, Nestor\}\]
n <- 3; s <- sample(U,size=n,replace=FALSE); s## [1] "Raul" "Nayibe" "Jhon"\[s=\{Raul, Nayibe, Jhon\}\]
n <- 4; s <- sample(U,size=n,replace=FALSE); s## [1] "Santiago" "Nayibe" "Raul" "Jhon"\[s=\{Santiago, Nayibe, Raul, Jhon\}\]
n <- 5; s <- sample(U,size=n,replace=FALSE); s## [1] "Nestor" "Jhon" "Santiago" "Nayibe" "Raul"\[s=\{Nestor, Jhon, Santiago, Nayibe, Raul\}\]
4.2 Muestra aleatoria con reemplazo
\[\boldsymbol{s}=\left(s_1,s_2,\ldots,s_k,\ldots,s_N\right)^t{\in}\mathbb{N}^{N}\]
\[n(S)={\sum}_{k{=}1}^{m}1\]
n <- 0; s <- sample(U,size=n,replace=TRUE); s## character(0)\[s=\{\}\]
n <- 1; s <- sample(U,size=n,replace=TRUE); s## [1] "Nestor"\[s=\{Nestor\}\]
n <- 2; s <- sample(U,size=n,replace=TRUE); s## [1] "Nayibe" "Nayibe"\[s=\{Nayibe, Nayibe\}\]
n <- 3; s <- sample(U,size=n,replace=TRUE); s## [1] "Nestor" "Santiago" "Jhon"\[s=\{Nestor, Santiago, Jhon\}\]
n <- 4; s <- sample(U,size=n,replace=TRUE); s## [1] "Jhon" "Raul" "Raul" "Nayibe"\[s=\{Jhon, Raul, Raul, Nayibe\}\]
n <- 5; s <- sample(U,size=n,replace=TRUE); s## [1] "Jhon" "Nestor" "Jhon" "Nestor" "Raul"\[s=\{Jhon, Nestor, Jhon, Nestor, Raul\}\]
n <- 6; s <- sample(U,size=n,replace=TRUE); s## [1] "Nayibe" "Jhon" "Santiago" "Santiago" "Jhon" "Nayibe"\[s=\{Nayibe, Jhon, Santiago, Santiago, Jhon, Nayibe\}\]
5 Soportes de muestreo
\[Q{\subseteq}\left\{{\{\},\{e_1\},\{e_2\},\ldots,\{e_k\},\ldots,\{e_N\},\{e_1,e_1\},\{e_1,e_2\},\ldots,\{e_1,e_k\},\ldots,\{e_1,e_N\},\ldots,\{e_1,e_2,\ldots,e_k,\ldots,e_N\}}\right\}\]
\[Q{\subseteq}\left\{{\{\},\{Santiago\},\{Nestor\},\ldots,\{Jhon\},\ldots,\{Santiago,Nestor,\ldots,Jhon\}}\right\}\]
5.1 Simétricos
\[s{\in}Q{\implies}{\sigma}{(s)}{\in}Q\text{; }{\sigma}(s)\text{: cualquier pernutación de }s\]
5.1.1 Sin reemplazo
\[\mathcal{S}=\{0,1\}^{N}\]
\[\#{\left(\mathcal{S}\right)}={2}^{N}\]
\(N=5\)
S <- expand.grid(c(0,1),c(0,1),c(0,1),c(0,1),c(0,1));colnames(S)<-U;S## Santiago Nestor Nayibe Raul Jhon
## 1 0 0 0 0 0
## 2 1 0 0 0 0
## 3 0 1 0 0 0
## 4 1 1 0 0 0
## 5 0 0 1 0 0
## 6 1 0 1 0 0
## 7 0 1 1 0 0
## 8 1 1 1 0 0
## 9 0 0 0 1 0
## 10 1 0 0 1 0
## 11 0 1 0 1 0
## 12 1 1 0 1 0
## 13 0 0 1 1 0
## 14 1 0 1 1 0
## 15 0 1 1 1 0
## 16 1 1 1 1 0
## 17 0 0 0 0 1
## 18 1 0 0 0 1
## 19 0 1 0 0 1
## 20 1 1 0 0 1
## 21 0 0 1 0 1
## 22 1 0 1 0 1
## 23 0 1 1 0 1
## 24 1 1 1 0 1
## 25 0 0 0 1 1
## 26 1 0 0 1 1
## 27 0 1 0 1 1
## 28 1 1 0 1 1
## 29 0 0 1 1 1
## 30 1 0 1 1 1
## 31 0 1 1 1 1
## 32 1 1 1 1 1N <- 5;cbind("N"=N, "#(s)"=2**(N))## N #(s)
## [1,] 5 32\[\mathcal{S}=\left\{(0, 0, 0, 0, 0),(1, 0, 0, 0, 0),\ldots,(1, 1, 0, 0, 0),\ldots,(1, 1, 1, 0, 0),\ldots,(1, 1, 1, 1, 0),\ldots,(1, 1, 1, 0, 1),\ldots,(1, 1, 1, 1, 1)\right\}\]
\[\#{\left(\mathcal{S}\right)}={2}^{5}=32\]
5.1.1.1 De tamaño fijo
\[\mathcal{S}_n=\left\{\boldsymbol{s}{\in}\mathcal{S}{\mid}{\sum}_{k{\in}U}s_k=n\right\}\]
\[\#{\left(\mathcal{S}\right)}=\binom{N}{n}=\binom{N}{N-n}\text{; }n=0,1,\ldots,N\]
\(N=5\text{ & }n=2\)
S <- S[rowSums(S)==2,];S## Santiago Nestor Nayibe Raul Jhon
## 4 1 1 0 0 0
## 6 1 0 1 0 0
## 7 0 1 1 0 0
## 10 1 0 0 1 0
## 11 0 1 0 1 0
## 13 0 0 1 1 0
## 18 1 0 0 0 1
## 19 0 1 0 0 1
## 21 0 0 1 0 1
## 25 0 0 0 1 1N <- 5;n <- 2;cbind("n"=n, "#(s)"=choose(N,n))## n #(s)
## [1,] 2 10\[\mathcal{S}_{2}=\left\{\boldsymbol{s}{\in}\mathcal{S}{\mid}{\sum}_{k{\in}U}s_k=2\right\}\]
\[\#{\left(\mathcal{S}_{2}\right)}=\binom{5}{2}=\binom{5}{5-2}=10\]
Support(N=length(U), n=2, U)## [,1] [,2]
## [1,] "Santiago" "Nestor"
## [2,] "Santiago" "Nayibe"
## [3,] "Santiago" "Raul"
## [4,] "Santiago" "Jhon"
## [5,] "Nestor" "Nayibe"
## [6,] "Nestor" "Raul"
## [7,] "Nestor" "Jhon"
## [8,] "Nayibe" "Raul"
## [9,] "Nayibe" "Jhon"
## [10,] "Raul" "Jhon"5.1.2 Con reemplazo
\[\mathcal{R}=\mathbb{N}^{N}\]
\[\#{\left(\mathcal{R}\right)}=+{\infty}\]
5.1.2.1 De tamaño fijo
\[\mathcal{R}_m=\left\{\boldsymbol{s}{\in}\mathcal{R}{\mid}{\sum}_{k{\in}U}s_k=m\right\}\]
\[\#{\left(\mathcal{R}_m\right)}=\binom{N+m-1}{m}=\binom{N+m-1}{N-1}\text{; }m=0,1,\ldots,N+m-1\]
\(N=5\text{ & }m=2\)
S <- rbind(S[rowSums(S)==2,],c(2,0,0,0,0),c(0,2,0,0,0),c(0,0,2,0,0),c(0,0,0,2,0),c(0,0,0,0,2));S## Santiago Nestor Nayibe Raul Jhon
## 4 1 1 0 0 0
## 6 1 0 1 0 0
## 7 0 1 1 0 0
## 10 1 0 0 1 0
## 11 0 1 0 1 0
## 13 0 0 1 1 0
## 18 1 0 0 0 1
## 19 0 1 0 0 1
## 21 0 0 1 0 1
## 25 0 0 0 1 1
## 111 2 0 0 0 0
## 12 0 2 0 0 0
## 131 0 0 2 0 0
## 14 0 0 0 2 0
## 15 0 0 0 0 2N <- 5;m <- 2;cbind("m"=m, "#(s)"=choose(N+m-1,m))## m #(s)
## [1,] 2 15\[\mathcal{R}_2=\left\{\boldsymbol{s}{\in}\mathcal{R}{\mid}{\sum}_{k{\in}U}s_k=2\right\}\]
\[\#{\left(\mathcal{R}_2\right)}=\binom{5+2-1}{2}=\binom{5+2-1}{5-1}=15\]
5.1.2.1.1 Interpretación: dos caramelos para repartir entre cinco niños
\[ \begin{bmatrix} Santiago & Santiago \\ Santiago & Nestor \\ \vdots & \vdots \\ Jhon & Jhon \end{bmatrix}{\implies} \begin{bmatrix} * & * & \mid{|} & \mid{|} & \mid{|} & \mid{|} \\ * & \mid{|} & * & \mid{|} & \mid{|} & \mid{|} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \mid{|} & \mid{|} & \mid{|}& \mid{|} & * & * \end{bmatrix} \]
5.1.2.1.2 Interpretación: dos asteriscos separados por cuatro barras
\[ \sigma\left[(*,*,\mid{|},\mid{|},\mid{|},\mid{|})\right]= \begin{bmatrix} * & * & \mid{|} & \mid{|} & \mid{|} & \mid{|} \\ * & \mid{|} & * & \mid{|} & \mid{|} & \mid{|} \\ * & \mid{|} & \mid{|} & * & \mid{|} & \mid{|} \\ * & \mid{|} & \mid{|} & \mid{|} & * & \mid{|} \\ * & \mid{|} & \mid{|} & \mid{|} & \mid{|} & * \\ \mid{|} & * & * & \mid{|} & \mid{|} & \mid{|} \\ \mid{|} & * & \mid{|} & * & \mid{|} & \mid{|} \\ \mid{|} & * & \mid{|} & \mid{|} & * & \mid{|} \\ \mid{|} & * & \mid{|} & \mid{|} & \mid{|} & * \\ \mid{|} & \mid{|} & * & * & \mid{|} & \mid{|} \\ \mid{|} & \mid{|} & * & \mid{|} & * & \mid{|} \\ \mid{|} & \mid{|} & * & \mid{|} & \mid{|} & * \\ \mid{|} & \mid{|} & \mid{|} & * & * & \mid{|} \\ \mid{|} & \mid{|} & \mid{|} & * & \mid{|} & * \\ \mid{|} & \mid{|} & \mid{|} & \mid{|} & * & * \end{bmatrix} \]
5.1.3 Propiedades
\(\mathcal{S}\), \(\mathcal{S}_{n}\), \(\mathcal{R}\) y \(\mathcal{R}_{m}\) son soportes simétricos
\(\mathcal{S}{\subset}\mathcal{R}\)
El conjunto \(\left\{\mathcal{S}_{0},\mathcal{S}_{1},\ldots,\mathcal{S}_{N}\right\}\) es una partición de \(\mathcal{S}\)
El conjunto \(\left\{\mathcal{R}_{0},\mathcal{R}_{1},\ldots,\mathcal{R}_{N},\ldots\right\}\) es una partición infinita de \(\mathcal{R}\)
\(\mathcal{S}_{n}{\subset}\mathcal{R}_{m}\) para todo \(n=m\) con \(n=0,1,\ldots,N\)
5.2 Muestras probabilísticas
Es posible construir o al menos definir teóricamente un soporte \(Q=\{s_1,\ldots,s_q,\ldots,s_Q\}\) de todas las muestras poaibles selecciondas por un método de selección en partícular; en donde \(s_{q=1,\ldots,Q}\) es una muestra dentro de \(Q\)
Las probabilidades de selección que el proceso aleatorio le otorga a cada una de las posibles muestras pertenecientes a \(Q\) son conocidas con antelación a la selección a la muestra seleccionada.
Ante la imposibilidad de construir un marco muestral es imposible llevar a cabo un muestreo del tipo probabilístico, y ante ello es inviable construir estimación alguna de tipo probabilístico, es decir, obtener confiabilidades, errores de muestreo y coeficientes de variación estimados
El científico de datos deberá responder por los engaños o fraudes, que por ignorancia, o mala fé o por la comodidad de mantener un empleo, negocio para el cual no está capacitado, viene cometiendo contra clientes, ciudadanos y países que confían en las cifras resultantes (Andrés Gutíerrez)
6 Diseño de muestreo
\[\mathcal{p}(\cdot):{Q}\rightarrow[0,1]\]
\[{\sum}_{s{\in}Q}\mathcal{p}(s)=1\]
\[{\forall}_{s{\in}Q}\mathcal{P}(S=s)=\mathcal{p}(s)\]
p <- c(.13, .2, .15, .1, .15, .04, .02, .06, .07, .08);p## [1] 0.13 0.20 0.15 0.10 0.15 0.04 0.02 0.06 0.07 0.086.1 Propiedades
\({\forall}_{s{\in}Q}\mathcal{p}(s){\geq}0\)
\({\sum}_{s{\in}Q}\mathcal{p}(s)=1\)
p > 0 | p == 0## [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUEp >= 0## [1] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUEsum(p)## [1] 16.2 Tipos
Sin reemplazo
Con reemplazo
De tamaño fijo
6.3 Algoritmo de selección
Procedimiento usado para seleccionar una muestra probabilística
6.3.1 Enumerativos
Listar todas la posibles muestras, generar una variable aleatoria con distribución uniforme en el intervalo \([0,1]\) y luego hacer la selección
6.4 Probabilidades de inclusión
\[ {I}_{k}(s)= \begin{cases} 1&\text{ si }k{\in}s\\ 0&\text{ si }k{\notin}s \end{cases} \]
6.4.1 De primer orden
\[ \begin{align} \pi_k&=\mathcal{P}(k{\in}s)\\ &=\mathcal{P}(I_k(s)=1)\\ &={\sum}_{s{\ni}k}\mathcal{p}(s) \end{align} \]
\[\pi_k={\sum}_{s{\ni}k}\mathcal{p}(s){\implies}\forall_{k{\in}U}\pi_k>0\]
\[\boldsymbol{\mu}=E(S)={\sum}_{s{\in}Q}s{\cdot}\mathcal{p}(S=s)\]
\[ \boldsymbol{\mu}= \begin{bmatrix} {\pi}_{1}\\ {\pi}_{2}\\ \vdots\\ {\pi}_{N} \end{bmatrix} =\boldsymbol{\pi} \]
Ik <- c(0,1)
design <- c(0.08,0.07,0.06,0.02,0.04,0.15,0.10,0.15,0.20,0.13)
mu <- t(expand.grid(Ik,Ik,Ik,Ik,Ik)[rowSums(expand.grid(Ik,Ik,Ik,Ik,Ik))==2,])%*%design
rownames(mu) <- c("Jhon","Raul","Nayibe","Nestor","Santiago")
colnames(mu) <- c("pik")
mu## pik
## Jhon 0.27
## Raul 0.33
## Nayibe 0.48
## Nestor 0.34
## Santiago 0.58\[\pi_k=E[I_k(S)]={\sum}_{s{\in}Q}I_k(s){\cdot}\mathcal{p}(s)\]
\[\pi_{Santiago}=E\left[I_{Santiago}(S)\right]={\sum}_{s{\in}Q}I_{Santiago}(s){\cdot}\mathcal{p}(s)=0.13+0.20+0.15+0.10=0.58\]
\[\pi_{Nestor}=E\left[I_{Nestor}(S)\right]={\sum}_{s{\in}Q}I_{Nestor}(s){\cdot}\mathcal{p}(s)=0.13+0.15+0.04+0.02=0.34\]
\[\pi_{Nayibe}=E\left[I_{Nayibe}(S)\right]={\sum}_{s{\in}Q}I_{Nayibe}(s){\cdot}\mathcal{p}(s)=0.20+0.15+0.06+0.07=0.48\]
\[\pi_{Raul}=E\left[I_{Raul}(S)\right]={\sum}_{s{\in}Q}I_{Raul}(s){\cdot}\mathcal{p}(s)=0.15+0.04+0.06+0.08=0.33\]
\[\pi_{Jhon}=E\left[I_{Jhon}(S)\right]={\sum}_{s{\in}Q}I_{Jhon}(s){\cdot}\mathcal{p}(s)=0.1+0.02+0.07+0.08=0.27\]
\[E(\pi_k)=n(s)\]
Ind <- Ik(N=5,n=2)
Ind## [,1] [,2] [,3] [,4] [,5]
## [1,] 1 1 0 0 0
## [2,] 1 0 1 0 0
## [3,] 1 0 0 1 0
## [4,] 1 0 0 0 1
## [5,] 0 1 1 0 0
## [6,] 0 1 0 1 0
## [7,] 0 1 0 0 1
## [8,] 0 0 1 1 0
## [9,] 0 0 1 0 1
## [10,] 0 0 0 1 1colnames(Ind) <- c("Santiago", "Nestor", "Nayibe", "Raul", "Jhon")
Ind## Santiago Nestor Nayibe Raul Jhon
## [1,] 1 1 0 0 0
## [2,] 1 0 1 0 0
## [3,] 1 0 0 1 0
## [4,] 1 0 0 0 1
## [5,] 0 1 1 0 0
## [6,] 0 1 0 1 0
## [7,] 0 1 0 0 1
## [8,] 0 0 1 1 0
## [9,] 0 0 1 0 1
## [10,] 0 0 0 1 1Q <- Support(N=5,n=2,U)
Q## [,1] [,2]
## [1,] "Santiago" "Nestor"
## [2,] "Santiago" "Nayibe"
## [3,] "Santiago" "Raul"
## [4,] "Santiago" "Jhon"
## [5,] "Nestor" "Nayibe"
## [6,] "Nestor" "Raul"
## [7,] "Nestor" "Jhon"
## [8,] "Nayibe" "Raul"
## [9,] "Nayibe" "Jhon"
## [10,] "Raul" "Jhon"data.frame(Q,p,Ind)## X1 X2 p Santiago Nestor Nayibe Raul Jhon
## 1 Santiago Nestor 0.13 1 1 0 0 0
## 2 Santiago Nayibe 0.20 1 0 1 0 0
## 3 Santiago Raul 0.15 1 0 0 1 0
## 4 Santiago Jhon 0.10 1 0 0 0 1
## 5 Nestor Nayibe 0.15 0 1 1 0 0
## 6 Nestor Raul 0.04 0 1 0 1 0
## 7 Nestor Jhon 0.02 0 1 0 0 1
## 8 Nayibe Raul 0.06 0 0 1 1 0
## 9 Nayibe Jhon 0.07 0 0 1 0 1
## 10 Raul Jhon 0.08 0 0 0 1 1\[\mu=E(S)={\sum}_{s{\in}Q}s{\cdot}\mathcal{p}(S=s)=\left(\pi_1,\pi_2\ldots,\pi_N\right)^t=\boldsymbol{\pi}\]
\[ \begin{align} \boldsymbol{\pi}&=\begin{pmatrix} 0.13 & 0.20 & 0.15 & 0.10 & 0.15 & 0.04 & 0.02 & 0.06 & 0.07 & 0.18 \end{pmatrix}\begin{pmatrix} I_{_{Santiago}}(s) & I_{_{Nestor}}(s) & 0 & 0 & 0\\ I_{_{Santiago}}(s) & 0 & I_{_{Nayibe}}(s) & 0 & 0\\ I_{_{Santiago}}(s) & 0 & 0 & I_{_{Raul}}(s) & 0\\ I_{_{Santiago}}(s) & 0 & 0 & 0 & I_{_{Jhon}}(s)\\ 0 & I_{_{Nestor}}(s) & I_{_{Nayibe}}(s) & 0 & 0\\ 0 & I_{_{Nestor}}(s) & 0 & I_{_{Raul}}(s) & 0\\ 0 & I_{_{Nestor}}(s) & 0 & 0 & I_{_{Jhon}}(s)\\ 0 & 0 & I_{_{Nayibe}}(s) & I_{_{Raul}}(s) & 0\\ 0 & 0 & I_{_{Nayibe}}(s) & 0 & I_{_{Jhon}}(s)\\ 0 & 0 & 0 & I_{_{Raul}}(s) & I_{_{Jhon}}(s) \end{pmatrix}\\ \end{align} \]
multip <- p*Ind
multip## Santiago Nestor Nayibe Raul Jhon
## [1,] 0.13 0.13 0.00 0.00 0.00
## [2,] 0.20 0.00 0.20 0.00 0.00
## [3,] 0.15 0.00 0.00 0.15 0.00
## [4,] 0.10 0.00 0.00 0.00 0.10
## [5,] 0.00 0.15 0.15 0.00 0.00
## [6,] 0.00 0.04 0.00 0.04 0.00
## [7,] 0.00 0.02 0.00 0.00 0.02
## [8,] 0.00 0.00 0.06 0.06 0.00
## [9,] 0.00 0.00 0.07 0.00 0.07
## [10,] 0.00 0.00 0.00 0.08 0.08colSums(multip)## Santiago Nestor Nayibe Raul Jhon
## 0.58 0.34 0.48 0.33 0.27pik <- Pik(p, Ind)
rownames(pik) <- c("pik")
pik## Santiago Nestor Nayibe Raul Jhon
## pik 0.58 0.34 0.48 0.33 0.27\[ \begin{align} \forall_{\mathcal{p}(\cdot)}n(s)={n}{\implies}E[n(s)]&={\sum}_U\pi_k\\ &=n \end{align} \]
sum(pik)## [1] 2pik > 0## Santiago Nestor Nayibe Raul Jhon
## pik TRUE TRUE TRUE TRUE TRUE6.4.2 De segundo orden
\[\pi_{kl}=E[I_k(S)\&I_l(S)]={\sum}_{s{\in}Q}I_{kl}(s){\cdot}\mathcal{p}(s)\]
\[\pi_{Santiago{\&}Nestor}=E\left[I_{Santiago}(S)\&I_{Nestor}(S)\right]={\sum}_{s{\in}Q}I_{Santiago{\&}Nestor}(s){\cdot}\mathcal{p}(s)=0.13\]
\[\pi_{Santiago{\&}Nayibe}=E\left[I_{Santiago}(S)\&I_{Nayibe}(S)\right]={\sum}_{s{\in}Q}I_{Santiago{\&}Nayibe}(s){\cdot}\mathcal{p}(s)=0.20\]
\[\pi_{Santiago{\&}Raul}=E\left[I_{Santiago}(S)\&I_{Raul}(S)\right]={\sum}_{s{\in}Q}I_{Santiago{\&}Raul}(s){\cdot}\mathcal{p}(s)=0.15\]
\[\pi_{Santiago{\&}Jhon}=E\left[I_{Santiago}(S)\&I_{Jhon}(S)\right]={\sum}_{s{\in}Q}I_{Santiago{\&}Jhon}(s){\cdot}\mathcal{p}(s)=0.10\]
\[\pi_{Nestor{\&}Nayibe}=E\left[I_{Nestor}(S)\&I_{Nayibe}(S)\right]={\sum}_{s{\in}Q}I_{Nestor{\&}Nayibe}(s){\cdot}\mathcal{p}(s)=0.15\]
\[\pi_{Nestor{\&}Raul}=E\left[I_{Nestor}(S)\&I_{Raul}(S)\right]={\sum}_{s{\in}Q}I_{Nestor{\&}Raul}(s){\cdot}\mathcal{p}(s)=0.04\]
\[\pi_{Nestor{\&}Jhon}=E\left[I_{Nestor}(S)\&I_{Jhon}(S)\right]={\sum}_{s{\in}Q}I_{Nestor{\&}Jhon}(s){\cdot}\mathcal{p}(s)=0.02\]
\[\pi_{Nayibe{\&}Raul}=E\left[I_{Nayibe}(S)\&I_{Raul}(S)\right]={\sum}_{s{\in}Q}I_{Nayibe{\&}Raul}(s){\cdot}\mathcal{p}(s)=0.06\]
\[\pi_{Nayibe{\&}Jhon}=E\left[I_{Nayibe}(S)\&I_{Jhon}(S)\right]={\sum}_{s{\in}Q}I_{Nayibe{\&}Jhon}(s){\cdot}\mathcal{p}(s)=0.07\]
\[\pi_{Raul{\&}Jhon}=E\left[I_{Raul}(S)\&I_{Jhon}(S)\right]={\sum}_{s{\in}Q}I_{Raul{\&}Jhon}(s){\cdot}\mathcal{p}(s)=0.08\]
data.frame(Q,Ind,p)## X1 X2 Santiago Nestor Nayibe Raul Jhon p
## 1 Santiago Nestor 1 1 0 0 0 0.13
## 2 Santiago Nayibe 1 0 1 0 0 0.20
## 3 Santiago Raul 1 0 0 1 0 0.15
## 4 Santiago Jhon 1 0 0 0 1 0.10
## 5 Nestor Nayibe 0 1 1 0 0 0.15
## 6 Nestor Raul 0 1 0 1 0 0.04
## 7 Nestor Jhon 0 1 0 0 1 0.02
## 8 Nayibe Raul 0 0 1 1 0 0.06
## 9 Nayibe Jhon 0 0 1 0 1 0.07
## 10 Raul Jhon 0 0 0 1 1 0.086.5 Estimación del total
\[\hat{t}_{y\pi}={\sum}_{k{\ni}S}\frac{y_k}{\pi_k}\]
y <- as.matrix(c(32,442,5454,646,656),col=1)
#y <- as.matrix(c(32,34,46,89,35),col=1)
colnames(y) <- c("yk")
rownames(y) <- c("Santiago", "Nestor", "Nayibe", "Raul", "Jhon")
y## yk
## Santiago 32
## Nestor 442
## Nayibe 5454
## Raul 646
## Jhon 656y["Santiago",]## [1] 32y["Nestor",]## [1] 442y["Nayibe",]## [1] 5454y["Raul",]## [1] 646y["Jhon",]## [1] 6566.5.1 \(\left\{Santiago,Nestor\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Santiago,Nestor\right\}}\frac{y_k}{\pi_k}=\frac{32}{0.58}+\frac{442}{0.34}\]
sum(y[c("Santiago","Nestor"),]/pik[,c("Santiago","Nestor")])## [1] 1355.1726.5.2 \(\left\{Santiago,Nayibe\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Santiago,Nayibe\right\}}\frac{y_k}{\pi_k}=\frac{32}{0.58}+\frac{5454}{0.48}\]
sum(y[c("Santiago","Nayibe"),]/pik[,c("Santiago","Nayibe")])## [1] 11417.676.5.3 \(\left\{Santiago,Raul\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Santiago,Raul\right\}}\frac{y_k}{\pi_k}=\frac{32}{0.58}+\frac{646}{0.33}\]
sum(y[c("Santiago","Raul"),]/pik[,c("Santiago","Raul")])## [1] 2012.7486.5.4 \(\left\{Santiago,Jhon\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Santiago,Jhon\right\}}\frac{y_k}{\pi_k}=\frac{32}{0.58}+\frac{656}{0.27}\]
sum(y[c("Santiago","Jhon"),]/pik[,c("Santiago","Jhon")])## [1] 2484.8026.5.5 \(\left\{Nestor,Nayibe\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Nestor,Nayibe\right\}}\frac{y_k}{\pi_k}=\frac{442}{0.34}+\frac{5454}{0.48}\]
sum(y[c("Nestor","Nayibe"),]/pik[,c("Nestor","Nayibe")])## [1] 12662.56.5.6 \(\left\{Nestor,Raul\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Nestor,Raul\right\}}\frac{y_k}{\pi_k}=\frac{442}{0.34}+\frac{646}{0.33}\]
sum(y[c("Nestor","Raul"),]/pik[,c("Nestor","Raul")])## [1] 3257.5766.5.7 \(\left\{Nestor,Jhon\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Nestor,Jhon\right\}}\frac{y_k}{\pi_k}=\frac{442}{0.34}+\frac{656}{0.27}\]
sum(y[c("Nestor","Jhon"),]/pik[,c("Nestor","Jhon")])## [1] 3729.636.5.8 \(\left\{Nayibe,Raul\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Nayibe,Raul\right\}}\frac{y_k}{\pi_k}=\frac{5454}{0.48}+\frac{646}{0.33}\]
sum(y[c("Nayibe","Raul"),]/pik[,c("Nayibe","Raul")])## [1] 13320.086.5.9 \(\left\{Nayibe,Jhon\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Nayibe,Jhon\right\}}\frac{y_k}{\pi_k}=\frac{5454}{0.48}+\frac{656}{0.27}\]
sum(y[c("Nayibe","Jhon"),]/pik[,c("Nayibe","Jhon")])## [1] 13792.136.5.10 \(\left\{Raul,Jhon\right\}\)
\[\hat{t}_{y}={\sum}_{\left\{Raul,Jhon\right\}}\frac{y_k}{\pi_k}=\frac{646}{0.33}+\frac{656}{0.27}\]
sum(y[c("Raul","Jhon"),]/pik[,c("Raul","Jhon")])## [1] 4387.2057 Características de interés y parámetros de interés
\[\mathbf{T}=\mathcal{f}(y_1,y_2,\ldots,y_n)\]
7.1 Total poblacional
\[t_y={\sum}_{k{\in}U}y_k\]
ty <- sum(y)
names(ty) <- "ty"
ty## ty
## 72307.1.1 Estimación
typi <- (Ind)%*%t(t(y)/pik)
colnames(typi) <- c("typi")
typi## typi
## [1,] 1355.172
## [2,] 11417.672
## [3,] 2012.748
## [4,] 2484.802
## [5,] 12662.500
## [6,] 3257.576
## [7,] 3729.630
## [8,] 13320.076
## [9,] 13792.130
## [10,] 4387.205data.frame(Q,p,Ind,typi)## X1 X2 p Santiago Nestor Nayibe Raul Jhon typi
## 1 Santiago Nestor 0.13 1 1 0 0 0 1355.172
## 2 Santiago Nayibe 0.20 1 0 1 0 0 11417.672
## 3 Santiago Raul 0.15 1 0 0 1 0 2012.748
## 4 Santiago Jhon 0.10 1 0 0 0 1 2484.802
## 5 Nestor Nayibe 0.15 0 1 1 0 0 12662.500
## 6 Nestor Raul 0.04 0 1 0 1 0 3257.576
## 7 Nestor Jhon 0.02 0 1 0 0 1 3729.630
## 8 Nayibe Raul 0.06 0 0 1 1 0 13320.076
## 9 Nayibe Jhon 0.07 0 0 1 0 1 13792.130
## 10 Raul Jhon 0.08 0 0 0 1 1 4387.2057.2 Media poblacional
\[\bar{y}_U=\frac{t_y}{N}=\frac{{\sum}_{k{\in}U}y_k}{N}\]
N <- nrow(y)
names(N) <- "N"
N## N
## 5ybar <- ty/N
names(ybar) <- "ybar"
ybar## ybar
## 14467.2.1 Estimación
ybars <- (1/n)*(Ind)%*%t(t(y)/pik)
colnames(typi) <- c("ybars")
ybars## yk
## [1,] 677.5862
## [2,] 5708.8362
## [3,] 1006.3741
## [4,] 1242.4010
## [5,] 6331.2500
## [6,] 1628.7879
## [7,] 1864.8148
## [8,] 6660.0379
## [9,] 6896.0648
## [10,] 2193.6027data.frame(Q,p,Ind,typi)## X1 X2 p Santiago Nestor Nayibe Raul Jhon ybars
## 1 Santiago Nestor 0.13 1 1 0 0 0 1355.172
## 2 Santiago Nayibe 0.20 1 0 1 0 0 11417.672
## 3 Santiago Raul 0.15 1 0 0 1 0 2012.748
## 4 Santiago Jhon 0.10 1 0 0 0 1 2484.802
## 5 Nestor Nayibe 0.15 0 1 1 0 0 12662.500
## 6 Nestor Raul 0.04 0 1 0 1 0 3257.576
## 7 Nestor Jhon 0.02 0 1 0 0 1 3729.630
## 8 Nayibe Raul 0.06 0 0 1 1 0 13320.076
## 9 Nayibe Jhon 0.07 0 0 1 0 1 13792.130
## 10 Raul Jhon 0.08 0 0 0 1 1 4387.2057.3 Varianza poblacional
\[S_{yU}^2=\frac{{\sum}_{k{\in}U}(y_k-\bar{y}_U)^2}{N-1}\]
SyU2 <- (t(y-ybar)%*%(y-ybar))/(N-1)
rownames(SyU2) <- "SyU2"
SyU2## yk
## SyU2 50838947.4 Estadística y estimador
\[T:Q\rightarrow\mathbb{R}\]
Cuando la estadística se usa para estimar un parámetro se dice estimador
A las realizaciones del estimador en una muestra \(s\) se les dice a.estimares
7.4.1 Esperanza
\[ E(T)={\sum}_{s{\in}Q}T(s){\cdot}\mathcal{p}(s) \]
7.4.2 Varianza
\[ \begin{align} V(T)&=E[T-E(T)]^2\\ &={\sum}_{s{\in}Q}[T(s)-E\left(T)\right]^2{\cdot}\mathcal{p}(s) \end{align} \]
7.4.3 Covarianza
\[ \begin{align} \forall_{T{\neq}U}C(T,U)&=E(T{\cdot}U)-E(T){\cdot}E(U)\\ &={\sum}_{s{\in}Q}\left[T(s){\cdot}U(s)\right]{\cdot}\mathcal{p}(s)-{\sum}_{s{\in}Q}T(s){\cdot}\mathcal{p}(s){\sum}_{s{\in}Q}U(s){\cdot}\mathcal{p}(s) \end{align} \]
7.4.4 \(I_k:S\rightarrow\left\{0,1\right\}\)
\[ \begin{align} I_k(s)&=\begin{cases} 1\text{ si }k{\in}S\\ 0\text{ si }k{\notin}S \end{cases} \end{align} \]
\[ \begin{align} E[I_k(s)]&=0{\cdot}\mathcal{p}[I_k(s)=0]+1{\cdot}\mathcal{p}[I_k(s)=1]\\ &=1{\cdot}\mathcal{p}(I_k=1)\\ &=\mathcal{p}(k{\in}S)\\ &=\pi_k \end{align} \]
\[ \begin{align} V(I_k(s))&=(0-\pi_k)^2{\cdot}\mathcal{p}[I_k(s)=0]+(1-\pi_k)^2{\cdot}\mathcal{p}[I_k(s)=1]\\ &=\pi_k^2(1-\pi_k)+(1-\pi_k)^2\pi_k\\ &=(1-\pi_k)[\pi_k^2+(1-\pi_k)\pi_k]\\ &=(1-\pi_k)(\pi_k^2+\pi_k-\pi_k^2)\\ &=(1-\pi_k)\pi_k\\ \end{align} \]
\[ \begin{align} V[I_{Santiago}(s)]&=(1-\pi_{Santiago})\pi_{Santiago}\\ &=(1-0.58)0.58\\ &=0.42{\cdot}0.58\\ &=0.2436 \end{align} \]
(1-0.58)*0.58## [1] 0.2436\[ \begin{align} V[I_{Nestor}(s)]&=(1-\pi_{Nestor})\pi_{Nestor}\\ &=(1-0.34)0.34\\ &=0.66{\cdot}0.34\\ &=0.2244 \end{align} \]
(1-0.34)*0.34## [1] 0.2244\[ \begin{align} V[I_{Nayibe}(s)]&=(1-\pi_{Nayibe})\pi_{Nayibe}\\ &=(1-0.48)0.48\\ &=0.52{\cdot}0.48\\ &=0.2496 \end{align} \]
(1-0.48)*0.48## [1] 0.2496\[ \begin{align} V[I_{Raul}(s)]&=(1-\pi_{Raul})\pi_{Raul}\\ &=(1-0.33)0.33\\ &=0.67{\cdot}0.33\\ &=0.2211 \end{align} \]
(1-0.33)*0.33## [1] 0.2211\[ \begin{align} V[I_{Jhon}(s)]&=(1-\pi_{Jhon})\pi_{Jhon}\\ &=(1-0.27)0.27\\ &=0.73{\cdot}0.27\\ &=0.1971 \end{align} \]
(1-0.27)*0.27## [1] 0.1971\[ \begin{align} \forall_{k{\neq}l}C(I_k(s),I_l(s))&=\left\{0{\cdot}[\mathcal{p}(I_{k}(s){\cdot}I_{l}(s)=0]+1{\cdot}\mathcal{p}[(I_{k}(s){\cdot}I_{l}(s)=1]\right\}-\pi_k{\cdot}\pi_l\\ &=\pi_{kl}-\pi_k{\cdot}\pi_l\\ &=\Delta_{kl} \end{align} \]
\[ \begin{align} C[I_{Santiago}(s),I_{Nestor}(s)]&=\pi_{Santiago,Nestor}-\pi_{Santiago}\pi_{Nestor}\\ &=0.13-0.58{\cdot}0.34\\ &=0.13-0.1972\\ &=-0.0672 \end{align} \]
0.13-0.58*0.34## [1] -0.0672\[ \begin{align} C[I_{Santiago}(s),I_{Nayibe}(s)]&=\pi_{Santiago,Nayibe}-\pi_{Santiago}\pi_{Nayibe}\\ &=0.20-0.58{\cdot}0.48\\ &=0.20-0.2784\\ &=-0.0784 \end{align} \]
0.20-0.58*0.48## [1] -0.0784\[ \begin{align} C[I_{Santiago}(s),I_{Raul}(s)]&=\pi_{Santiago,Raul}-\pi_{Santiago}\pi_{Raul}\\ &=0.15-0.58{\cdot}0.33\\ &=0.15-0.1914\\ &=-0.0414 \end{align} \]
0.15-0.58*0.33## [1] -0.0414\[ \begin{align} C[I_{Santiago}(s),I_{Jhon}(s)]&=\pi_{Santiago,Jhon}-\pi_{Santiago}\pi_{Jhon}\\ &=0.10-0.58{\cdot}0.27\\ &=0.10-0.1566\\ &=-0.0566 \end{align} \]
0.10-0.58*0.27## [1] -0.0566\[ \begin{align} C[I_{Nestor}(s),I_{Nayibe}(s)]&=\pi_{Nestor,Nayibe}-\pi_{Nestor}\pi_{Nayibe}\\ &=0.15-0.34{\cdot}0.48\\ &=0.15-0.1632\\ &=-0.0132 \end{align} \]
0.15-0.34*0.48## [1] -0.0132\[ \begin{align} C[I_{Nestor}(s),I_{Raul}(s)]&=\pi_{Nestor,Raul}-\pi_{Nestor}\pi_{Raul}\\ &=0.04-0.34{\cdot}0.33\\ &=0.04-0.1122\\ &=-0.0722 \end{align} \]
0.04-0.34*0.33## [1] -0.0722\[ \begin{align} C[I_{Nestor}(s),I_{Jhon}(s)]&=\pi_{Nestor,Jhon}-\pi_{Nestor}\pi_{Jhon}\\ &=0.02-0.34{\cdot}0.27\\ &=0.02-0.0918\\ &=-0.0718 \end{align} \]
0.02-0.34*0.27## [1] -0.0718\[ \begin{align} C[I_{Nayibe}(s),I_{Raul}(s)]&=\pi_{Nayibe,Raul}-\pi_{Nayibe}\pi_{Raul}\\ &=0.06-0.48{\cdot}0.33\\ &=0.06-0.1584\\ &=-0.0984 \end{align} \]
0.06-0.48*0.33## [1] -0.0984\[ \begin{align} C[I_{Nayibe}(s),I_{Jhon}(s)]&=\pi_{Nayibe,Jhon}-\pi_{Nayibe}\pi_{Jhon}\\ &=0.07-0.48{\cdot}0.27\\ &=0.07-0.1296\\ &=-0.0596 \end{align} \]
0.07-0.48*0.27## [1] -0.0596\[ \begin{align} C[I_{Raul}(s),I_{Jhon}(s)]&=\pi_{Raul,Jhon}-\pi_{Raul}\pi_{Jhon}\\ &=0.08-0.33{\cdot}0.27\\ &=0.08-0.0891\\ &=-0.0091 \end{align} \]
0.08-0.33*0.27## [1] -0.00917.4.5 \(n(S):S\rightarrow\mathbb{Z}^{+}\)
\[ \begin{align} n(S)&={\sum}_{k{\in}U}I_k(s)\\ &={\sum}_{k{\in}U}I_k \end{align} \]
\[ \begin{align} E[n(S)]&=E\left[{\sum}_{U}I_k(s)\right]\\ &={\sum}_{U}E\left[I_k(s)\right]\\ &={\sum}_{U}\pi_k \end{align} \]
\[ \begin{align} E[n(S)]&=\pi_{Santiago}+\pi_{Nestor}+\pi_{Nayibe}+\pi_{Raul}+\pi_{Jhon}\\ &=0.58+0.34+0.48+0.33+0.27\\ &=2 \end{align} \]
sum(pik)## [1] 2\[ \begin{align} V[n(S)]&=V\left[{\sum}_{U}I_k(s)\right]\\ &={\sum}_{k{\in}U}V\left[I_k(s)\right]+{\sum}_{k{\in}U}{\sum}_{k{\neq}l}C\left[I_k(s),I_l(s)\right]\\ &={\sum}_{k{\in}U}(\pi_k-\pi_k^2)+{\sum}_{k{\in}U}{\sum}_{k{\neq}l}(\pi_{kl}-\pi_k\pi_l)\\ &={\sum}_{k{\in}U}(\pi_k-\pi_k^2)-{\sum}_{k{\in}U}{\sum}_{k{\neq}l}(\pi_k\pi_l-\pi_{kl})\\ &={\sum}_{k{\in}U}\pi_k-{\sum}_{k{\in}U}\pi_k^2-{\sum}_{k{\in}U}{\sum}_{k{\neq}l}\pi_k\pi_l+{\sum}_{k{\in}U}{\sum}_{k{\neq}l}\pi_{kl}\\ &={\sum}_{k{\in}U}\pi_k-\left({\sum}_{k{\in}U}\pi_k\right)^2+{\sum}_{k{\in}U}{\sum}_{k{\neq}l}\pi_{kl} \end{align} \]
\[ \begin{align} \left({\sum}_{k{\in}U}\pi_k\right)^2&={\sum}_{k{\in}U}\pi_k{\sum}_{k{\in}U}\pi_k\\ &={\sum}_{k{\in}U}\pi_k^2+{\sum}_{k{\in}U}{\sum}_{k{\neq}l}\pi_k\pi_l\\ \end{align} \]
Sample <- Ik(N=5, n=2)
SubInd <- OrderWR(N=5, 2)
K <- matrix(c(Sample[, SubInd]), ncol=2)
L <- t(t(K[, 1]) * K[, 2])
Ikl <- matrix(c(L), ncol=nrow(SubInd))
M <- p * Ikl
O <- apply(M, 2, sum)
P <- matrix(c(O), ncol=N)
colnames(P) = rownames(P) <- c("Santiago", "Nestor", "Nayibe", "Raul", "Jhon")
P## Santiago Nestor Nayibe Raul Jhon
## Santiago 0.58 0.13 0.20 0.15 0.10
## Nestor 0.13 0.34 0.15 0.04 0.02
## Nayibe 0.20 0.15 0.48 0.06 0.07
## Raul 0.15 0.04 0.06 0.33 0.08
## Jhon 0.10 0.02 0.07 0.08 0.27library(TeachingSampling)
pikl <- Pikl(N=5, n=2, p=p)
colnames(pikl)=rownames(pikl) <- c("Santiago", "Nestor", "Nayibe", "Raul", "Jhon")
pikl## Santiago Nestor Nayibe Raul Jhon
## Santiago 0.58 0.13 0.20 0.15 0.10
## Nestor 0.13 0.34 0.15 0.04 0.02
## Nayibe 0.20 0.15 0.48 0.06 0.07
## Raul 0.15 0.04 0.06 0.33 0.08
## Jhon 0.10 0.02 0.07 0.08 0.27\[ \begin{align} V[n(S)]&=\pi_{Santiago}+\pi_{Nestor}+\pi_{Nayibe}+\pi_{Raul}+\pi_{Jhon}-\left(\pi_{Santiago}+\pi_{Nestor}+\pi_{Nayibe}+\pi_{Raul}+\pi_{Jhon}\right)^2+{\sum}_{k{\in}U}{\sum}_{k{\neq}l}\pi_{kl}\\ &=0.58+0.34+0.48+0.33+0.27-(0.58+0.34+0.48+0.33+0.27)^2+{\sum}_{k{\in}U}{\sum}_{k{\neq}l}\pi_{kl}\\ &=2-2^2+\pi_{Santiago,Nestor}+\pi_{Santiago,Nayibe}+\cdots+\pi_{Santiago,Jhon}+\cdots+\pi_{Raul,Jhon}\\ &=2-4+0.13+0.20+\cdots+0.10+\cdots+0.08\\ &=-2+\pi_{Santiago,Nestor}+\pi_{Santiago,Nayibe}+\cdots+\pi_{Santiago,Jhon}+\cdots+\pi_{Raul,Jhon}\\ &=-2+2\\ &=0 \end{align} \]
diag(pikl)=0
sum(pik)-sum(pik)**2+sum(rowSums(pikl))## [1] 0\[ \begin{align} {\sum}_{l{\in}U}\pi_{kl}&={\sum}_{l{\in}U}E[I_k(S)I_l(S)]\\ &={\sum}_{l{\in}U}{\sum}_{s{\in}Q}I_k(s)I_l(s)\mathcal{p}(s)\\ &={\sum}_{s{\in}Q}I_k(s)\mathcal{p}(s){\sum}_{l{\in}U}I_l(S)\\ &=n(S){\sum}_{s{\in}Q}I_l(s)\mathcal{p}(s)\\ &=n{\cdot}\pi_{l} \end{align} \]
data.frame(Q,Ind,p)## X1 X2 Santiago Nestor Nayibe Raul Jhon p
## 1 Santiago Nestor 1 1 0 0 0 0.13
## 2 Santiago Nayibe 1 0 1 0 0 0.20
## 3 Santiago Raul 1 0 0 1 0 0.15
## 4 Santiago Jhon 1 0 0 0 1 0.10
## 5 Nestor Nayibe 0 1 1 0 0 0.15
## 6 Nestor Raul 0 1 0 1 0 0.04
## 7 Nestor Jhon 0 1 0 0 1 0.02
## 8 Nayibe Raul 0 0 1 1 0 0.06
## 9 Nayibe Jhon 0 0 1 0 1 0.07
## 10 Raul Jhon 0 0 0 1 1 0.08\[{\sum}_U\pi_{Santiago,l}=\pi_{Santiago,Santiago}+\pi_{Nestor,Santiago}+\pi_{Nayibe,Santiago}+\pi_{Raul,Santiago}+\pi_{Jhon,Santiago}=n{\cdot}\pi_{Santiago}\]
all.equal(.58+.13+.15+.20+.10,2*.58)## [1] TRUE\[{\sum}_U\pi_{Nestor,l}=\pi_{Santiago,Nestor}+\pi_{Nestor,Nestor}+\pi_{Nayibe,Nestor}+\pi_{Raul,Nestor}+\pi_{Jhon,Nestor}=n{\cdot}\pi_{Nestor}\]
all.equal(.13+.34+.15+.04+.02,2*.34)## [1] TRUE\[{\sum}_U\pi_{Nayibe,l}=\pi_{Santiago,Nayibe}+\pi_{Nestor,Nayibe}+\pi_{Nayibe,Nayibe}+\pi_{Raul,Nayibe}+\pi_{Jhon,Nayibe}=n{\cdot}\pi_{Nayibe}\]
all.equal(.20+0.15+.48+.06+.07,2*.48)## [1] TRUE\[{\sum}_U\pi_{Raul,l}=\pi_{Santiago,Raul}+\pi_{Nestor,Raul}+\pi_{Nayibe,Raul}+\pi_{Jhon,Raul}+\pi_{Jhon,Raul}=n{\cdot}\pi_{Raul}\]
all.equal(.15+.04+.06+.33+.08,2*.33)## [1] TRUE\[{\sum}_U\pi_{Jhon,l}=\pi_{Santiago,Jhon}+\pi_{Nestor,Jhon}+\pi_{Nayibe,Jhon}+\pi_{Raul,Jhon}+\pi_{Jhon,Jhon}=n{\cdot}\pi_{Jhon}\]
all.equal(.10+.02+.07+.08+.27,2*.27)## [1] TRUEdata.frame(Q,Ind,p)## X1 X2 Santiago Nestor Nayibe Raul Jhon p
## 1 Santiago Nestor 1 1 0 0 0 0.13
## 2 Santiago Nayibe 1 0 1 0 0 0.20
## 3 Santiago Raul 1 0 0 1 0 0.15
## 4 Santiago Jhon 1 0 0 0 1 0.10
## 5 Nestor Nayibe 0 1 1 0 0 0.15
## 6 Nestor Raul 0 1 0 1 0 0.04
## 7 Nestor Jhon 0 1 0 0 1 0.02
## 8 Nayibe Raul 0 0 1 1 0 0.06
## 9 Nayibe Jhon 0 0 1 0 1 0.07
## 10 Raul Jhon 0 0 0 1 1 0.08\[ \begin{align} {\sum}_U\Delta_{kl}&={\sum}_{U}\left(\pi_{kl}-\pi_k\pi_l\right)\\ &={\sum}_{U}\pi_{kl}-{\sum}_{U}\pi_k\pi_l\\ &=n\pi_{l}-\pi_l{\sum}_{U}\pi_k\\ &=n\pi_{l}-\pi_ln\\ &=0 \end{align} \]
Ind <- Ik(N=5, n=2)
multip <- p * Ind
pik <- colSums(multip)
t(pik)## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.58 0.34 0.48 0.33 0.27P1 <- as.matrix(t(pik))
deltakl <- P - (t(P1) %*% P1)
sum(deltakl)## [1] -3.608225e-16deltakl <- Deltakl(N=5, n=2, p=p)
sum(deltakl)## [1] -3.608225e-16\[ \begin{align} \pi_k\left(1-\pi_k\right)&=V\left[I_{k}(S)\right]\\ &=C\left[I_{k}(S),I_{k}(S)\right]\\ &=C\left[I_{k}(S),n-{\sum}_{l{\neq}k}I_{l}(S)\right]\\ &=C\left[I_{k}(S),n\right]-{\sum}_{l{\neq}k}C\left[I_{k}(S),I_{l}(S)\right]\\ &=0-{\sum}_{l{\neq}k}C\left[I_{k}(S),I_{l}(S)\right]\\ &=-{\sum}_{l{\neq}k}(\pi_{kl}-\pi_{k}\pi_{l})\\ &={\sum}_{l{\neq}k}(\pi_{k}\pi_{l}-\pi_{kl})\\ \end{align} \]
pikl <- Pikl(N=5, n=2, p=p)
colnames(pikl)=rownames(pikl) <- c("Santiago", "Nestor", "Nayibe", "Raul", "Jhon")
pikl## Santiago Nestor Nayibe Raul Jhon
## Santiago 0.58 0.13 0.20 0.15 0.10
## Nestor 0.13 0.34 0.15 0.04 0.02
## Nayibe 0.20 0.15 0.48 0.06 0.07
## Raul 0.15 0.04 0.06 0.33 0.08
## Jhon 0.10 0.02 0.07 0.08 0.27\[ \begin{align} \pi_{Santiago}\left(1-\pi_{Santiago}\right)&={\sum}_{l{\neq}{Santiago}}(\pi_{{Santiago}}\pi_{l}-\pi_{{Santiago},l})\\ \end{align} \]
all.equal(pikl["Santiago","Santiago"]*(1-pikl["Santiago","Santiago"]),sum(pikl["Santiago","Santiago"]*pikl["Nestor","Nestor"]-pikl["Santiago","Nestor"],pikl["Santiago","Santiago"]*pikl["Nayibe","Nayibe"]-pikl["Santiago","Nayibe"],pikl["Santiago","Santiago"]*pikl["Raul","Raul"]-pikl["Santiago","Raul"],pikl["Santiago","Santiago"]*pikl["Jhon","Jhon"]-pikl["Santiago","Jhon"]))## [1] TRUE\[ \begin{align} \pi_{Nestor}\left(1-\pi_{Nestor}\right)&={\sum}_{l{\neq}{Nestor}}(\pi_{{Nestor}}\pi_{l}-\pi_{{Nestor},l})\\ \end{align} \]
all.equal(pikl["Nestor","Nestor"]*(1-pikl["Nestor","Nestor"]),sum(pikl["Nestor","Nestor"]*pikl["Santiago","Santiago"]-pikl["Nestor","Santiago"],pikl["Nestor","Nestor"]*pikl["Nayibe","Nayibe"]-pikl["Nestor","Nayibe"],pikl["Nestor","Nestor"]*pikl["Raul","Raul"]-pikl["Nestor","Raul"],pikl["Nestor","Nestor"]*pikl["Jhon","Jhon"]-pikl["Nestor","Jhon"]))## [1] TRUE\[ \begin{align} \pi_{Nayibe}\left(1-\pi_{Nayibe}\right)&={\sum}_{l{\neq}{Nayibe}}(\pi_{{Nayibe}}\pi_{l}-\pi_{{Nayibe},l})\\ \end{align} \]
all.equal(pikl["Nayibe","Nayibe"]*(1-pikl["Nayibe","Nayibe"]),sum(pikl["Nayibe","Nayibe"]*pikl["Santiago","Santiago"]-pikl["Nayibe","Santiago"],pikl["Nayibe","Nayibe"]*pikl["Nestor","Nestor"]-pikl["Nayibe","Nestor"],pikl["Nayibe","Nayibe"]*pikl["Raul","Raul"]-pikl["Nayibe","Raul"],pikl["Nayibe","Nayibe"]*pikl["Jhon","Jhon"]-pikl["Nayibe","Jhon"]))## [1] TRUE\[ \begin{align} \pi_{Raul}\left(1-\pi_{Raul}\right)&={\sum}_{l{\neq}{Raul}}(\pi_{{Raul}}\pi_{l}-\pi_{{Raul},l})\\ \end{align} \]
all.equal(pikl["Raul","Raul"]*(1-pikl["Raul","Raul"]),sum(pikl["Raul","Raul"]*pikl["Santiago","Santiago"]-pikl["Raul","Santiago"],pikl["Raul","Raul"]*pikl["Nestor","Nestor"]-pikl["Raul","Nestor"],pikl["Raul","Raul"]*pikl["Nayibe","Nayibe"]-pikl["Raul","Nayibe"],pikl["Raul","Raul"]*pikl["Jhon","Jhon"]-pikl["Raul","Jhon"]))## [1] TRUE\[ \begin{align} \pi_{Jhon}\left(1-\pi_{Jhon}\right)&={\sum}_{l{\neq}{Jhon}}(\pi_{{Jhon}}\pi_{l}-\pi_{{Jhon},l})\\ \end{align} \]
all.equal(pikl["Jhon","Jhon"]*(1-pikl["Jhon","Jhon"]),sum(pikl["Jhon","Jhon"]*pikl["Santiago","Santiago"]-pikl["Jhon","Santiago"],pikl["Jhon","Jhon"]*pikl["Nestor","Nestor"]-pikl["Jhon","Nestor"],pikl["Jhon","Jhon"]*pikl["Nayibe","Nayibe"]-pikl["Jhon","Nayibe"],pikl["Jhon","Jhon"]*pikl["Raul","Raul"]-pikl["Jhon","Raul"]))## [1] TRUE7.5 Propiedades de un estimador
- Sesgo
\[B(\widehat{T})=E(\widehat{T})-T\]
- Error cuadrático medio
\[ \begin{align} ECM(\widehat{T})&=E[\widehat{T}-T]^2\\ &=V(\widehat{T})+B^2(\widehat{T}) \end{align} \]
8 Estrategia de muestreo
\[\left(\mathcal{p}(\cdot),\widehat{T}\right)\text{ con }\widehat{T}\text{ un estimador de un parámetro }T\text{ y }\mathcal{p}(\cdot)\text{ un diseño de muestreo definido sobre un soporte }Q\]
8.1 Estimadores de muestreo
El objetivo de una investigación por muestreo probabilístico es estimar una parámetro (o valor poblacional fijo) de interés a través de una muestra aleatoria (o subconjunto de la población elegido siguiendo un diseño de muestreo estadístico). Una medida de precisión de un estimador es dado por:
- Indicador de precisión
\[cve(\widehat{T})=\frac{\sqrt{\widehat{V}(\widehat{T})}}{\widehat{T}}\]
Una medida comunmente usada para expresar el error cometido al seleccionar una muestra y no utilizar toda la población
8.1.1 Estimador de Horvitz - Thompson
\[ \begin{align} \widehat{t}_{y,\pi}&={\sum}_{s}\frac{y_k}{\pi_k}\\ &={\sum}_{s}f_ky_k \end{align} \]
8.1.1.1 Esperanza del estimador
\[ \begin{align} \forall_{k{\in}U}\pi_k>0{\implies}E(\widehat{t}_{y,\pi})&={t}_{y} \end{align} \]
\[ \begin{align} E[\widehat{t}_{y,\pi}]&=E\left[{\sum}_{s}\frac{y_k}{\pi_k}\right]\\ &=E\left[{\sum}_{U}I_k(S)\frac{y_k}{\pi_k}\right]\\ &={\sum}_{U}E\left[I_k(S)\right]\frac{y_k}{\pi_k}\\ &={\sum}_{U}\pi_k\frac{y_k}{\pi_k}\\ &={\sum}_{U}y_k\\ &=t_y \end{align} \]
8.1.1.2 Varianza del estimador
\[ \begin{align} V[\widehat{t}_{y,\pi}]&=V\left[{\sum}_{s}\frac{y_k}{\pi_k}\right]\\ &={\sum}_{U}V\left[I_k(S)\right]\frac{y_k^2}{\pi_k^2}+{\sum}_{U}{\sum}_{k{\neq}l}C\left[I_k(S),I_l(S)\right]\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}\\ &={\sum}_{U}\left[\pi_k-\pi_k^2\right]\frac{y_k^2}{\pi_k^2}+{\sum}_{U}{\sum}_{k{\neq}l}\left[\pi_{kl}-\pi_k\pi_l\right]\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}\\ &={{\sum}{\sum}}_{U}\left[\pi_{kl}-\pi_k\pi_l\right]\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}\\ &={{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l} \end{align} \]
\[ \begin{align} n(S)=n{\implies}V[\widehat{t}_{y,\pi}]&=-\frac{1}{2}{{\sum}{\sum}}_{U}\Delta_{kl}\left[\frac{y_k}{\pi_k}-\frac{y_l}{\pi_l}\right]^2 \end{align} \]
\[ \begin{align} V[\widehat{t}_{y,\pi}]&=-\frac{1}{2}{{\sum}{\sum}}_{U}\Delta_{kl}\left[\frac{y_k}{\pi_k}-\frac{y_l}{\pi_l}\right]^2\\ &=-\frac{1}{2}{{\sum}{\sum}}_{U}\Delta_{kl}\left[\frac{y_k^2}{\pi_k^2}-2\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}+\frac{y_l^2}{\pi_l^2}\right]\\ &=-\frac{1}{2}\left[{{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k^2}{\pi_k^2}-2{{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}+{{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_l^2}{\pi_l^2}\right]\\ &=-\frac{1}{2}\left[2{{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k^2}{\pi_k^2}-2{{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}\right]\\ &=-{\sum}_{k{\in}U}\frac{y_k^2}{\pi_k^2}{\sum}_{l{\in}U}\Delta_{kl}+{{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}\\ &=-\left[{\sum}_{k{\in}U}\frac{y_k^2}{\pi_k^2}\right]{\cdot}0+{{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}\\ &={{\sum}{\sum}}_{U}\Delta_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l} \end{align} \]
8.1.2 Estimador de la varianza
- Un estimador para \(\widehat{t}_{y,\pi}\)
\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&={{\sum}{\sum}}_{s}\frac{\Delta_{kl}}{\pi_{kl}}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l} \end{align} \]
- Si el diseño es de tamaño fijo \(n(S)=cte\)
\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=-\frac{1}{2}{{\sum}{\sum}}_{U}\frac{\Delta_{kl}}{\pi_{kl}}\left[\frac{y_k}{\pi_k}-\frac{y_l}{\pi_l}\right]^2 \end{align} \]
8.1.3 Intervalo de confianza
Suponiendo normalidad
\[{\sum}_{Q_0{\supset}s}\mathcal{p}(s)\stackrel{n{\rightarrow}\infty}{=}P\left(Z<z_{1-\alpha}\right)=1-\alpha\]
- En la teoría
\[\widehat{t}_{y,\pi}{\pm}z_{1-\frac{\alpha}{2}}\sqrt{V[\widehat{t}_{y,\pi}]}\]
- En la práctica
\[\widehat{t}_{y,\pi}{\pm}z_{1-\frac{\alpha}{2}}\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}\]
8.1.3.1 Confiabilidad
\[ {\sum}_{Q_0{\supset}s}\mathcal{p}\left(\widehat{t}_{y,\pi}-z_{1-\frac{\alpha}{2}}\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}<{t}_{y}<\widehat{t}_{y,\pi}+z_{1-\frac{\alpha}{2}}\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}\right)\stackrel{n{\rightarrow}\infty}{=}1-\alpha \]
8.1.3.2 Precisión
\[\left[\widehat{t}_{y,\pi}{+}z_{1-\frac{\alpha}{2}}\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}\right]-\left[\widehat{t}_{y,\pi}{-}z_{1-\frac{\alpha}{2}}\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}\right]=2{\times}z_{1-\frac{\alpha}{2}}\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}\]
8.1.4 Estimación de otros parámetros
\[ \begin{align} \widehat{\bar{y}}_{\pi}&=\frac{1}{N}\widehat{t}_{y,\pi}\\ &=\frac{1}{N}{\sum}_{s}\frac{y_k}{\pi_k}\\ \end{align} \]
\[ \begin{align} V\left[\widehat{\bar{y}}_{\pi}\right]&=V\left[\frac{1}{N}\widehat{t}_{y,\pi}\right]\\ &=\frac{1}{N^2}V\left[\widehat{t}_{y,\pi}\right]\\ \end{align} \]
\[ \begin{align} \widehat{V}\left[\widehat{\bar{y}}_{\pi}\right]&=\widehat{V}\left[\frac{1}{N}\widehat{t}_{y,\pi}\right]\\ &=\frac{1}{N^2}\widehat{V}\left[\widehat{t}_{y,\pi}\right]\\ \end{align} \]
\[ N={\sum}_{U}1 \]
\[ \widehat{N}_{\pi}={\sum}_{s}\frac{1}{\pi_{k}} \]
\[ \begin{align} \tilde{y}_{s}&=\frac{{\sum}_{s}\frac{y_k}{\pi_{k}}}{{\sum}_{s}\frac{1}{\pi_{k}}}\\ &=\frac{\widehat{t}_{y\pi}}{\widehat{N}_{\pi}} \end{align} \]
8.1.4.1 Aún cuando se conoce N
\[\forall_{k{\in}U}y_k=c\]
\[ \begin{align} \widehat{\bar{y}}_{\pi}&=\frac{1}{N}\widehat{t}_{y,\pi}\\ &=\frac{1}{N}{\sum}_{s}\frac{c}{\pi_k}\\ &=\frac{c}{N}{\sum}_{s}\frac{1}{\pi_k}\\ &=\frac{c}{N}\widehat{N}_{\pi}\\ &=c\frac{\widehat{N}_{\pi}}{N}\\ \end{align} \]
\[ \begin{align} \tilde{y}_{s}&=\frac{{\sum}_{s}\frac{c}{\pi_{k}}}{{\sum}_{s}\frac{1}{\pi_{k}}}\\ &=\frac{c{\sum}_{s}\frac{1}{\pi_{k}}}{{\sum}_{s}\frac{1}{\pi_{k}}}\\ &=\frac{c\widehat{N}_{\pi}}{\widehat{N}_{\pi}}\\ &=c \end{align} \]
8.1.5 Estimador alternativo
\[ \begin{align} \widehat{t}_{y,alt}&=N\tilde{y}_{s}\\ &=N\frac{{\sum}_{s}\frac{y_k}{\pi_{k}}}{{\sum}_{s}\frac{1}{\pi_{k}}}\\ &=N\frac{\widehat{t}_{y\pi}}{\widehat{N}_{\pi}}\\ &=\frac{N}{\widehat{N}_{\pi}}\widehat{t}_{y\pi} \end{align} \]
8.1.6 Estimador de Hansen - Hurtwitz
Diseño de muestreo con reemplazo; se extraen \(m\) muestras, de manera independiente, de tamaño \(1\)
\[m\text{ muestras independientes de tamaño }1\]
\[{\sum}_{U}p_k=1\]
\[p_k\text{: probabilidad de selección}\]
\[{\forall}_{k{\in}U\&i=1,\ldots,m}\mathcal{P}(k{\in}s_i)=p_k\]
El elemento seleccionado es reemplazado en la población y vuelve a ser parte del próximo sorteo aleatorio con la misma probabilidad de selección \(p_k\)
La probabilidad de inclusión \(\pi_k\) no es lo mismo que la probabilidad de selección \(p_k\)
\[\#(S)=n_{k}(S){\leq}m\]
\[E[n_{k}(S)]=mp_k\]
\[V[n_{k}(S)]=mp_k(1-p_k)\]
\[\pi_k\text{: es la probabilidad de que el elemento sea seleccionado al menos una vez en la muestra}\]
8.1.6.1 Diseño de muestreo (distribución multinomial)
\[ \mathcal{p}(s)= \begin{cases} \frac{m!}{n_1(s)!n_2(s)!{\cdots}n_N(s)!}{\prod}_{U}({p}_{k})^{n_k(s)} & \text{si }{\sum}_{U}n_k(s)=m\\ 0 & \text{en otro caso} \end{cases} \]
\[{\sum}_{x{\in}Q}\mathcal{p}(s)=1\]
8.1.6.2 Cardinalidad del soporte
\[\#(Q)=\binom{N+m-1}{m}\]
library(gtools)
combinations(n=5,r=2,v=U, set=TRUE, repeats.allowed=TRUE)## [,1] [,2]
## [1,] "Jhon" "Jhon"
## [2,] "Jhon" "Nayibe"
## [3,] "Jhon" "Nestor"
## [4,] "Jhon" "Raul"
## [5,] "Jhon" "Santiago"
## [6,] "Nayibe" "Nayibe"
## [7,] "Nayibe" "Nestor"
## [8,] "Nayibe" "Raul"
## [9,] "Nayibe" "Santiago"
## [10,] "Nestor" "Nestor"
## [11,] "Nestor" "Raul"
## [12,] "Nestor" "Santiago"
## [13,] "Raul" "Raul"
## [14,] "Raul" "Santiago"
## [15,] "Santiago" "Santiago"8.1.6.3 Probabilidades de inclusión
\[ \begin{align} \pi_k&=\mathcal{p}(k{\in}S)\\ &=1-\mathcal{p}(k{\notin}S)\\ &=1-\binom{m}{m}{(1-p_k)}^{m}{p_k}^{m-m}\\ &=1-{\left(1-p_k\right)}^{m} \end{align} \]
\[ \begin{align} \forall_{i=1,\ldots,m}\mathcal{P}(k,l{\not\in}s_i)&=\mathcal{P}(k{\not\in}s_i\&l{\not\in}s_i)\\ &=1-\mathcal{P}(k{\in}s_i{\cup}l{\in}s_i)\\ &=1-\left[\mathcal{P}(k{\in}s_i)+\mathcal{P}(l{\in}s_i)-\mathcal{P}(k{\in}s_i{\cap}l{\in}s_i)\right]\\ &=1-\left[p_k+p_l\right]\\ &=1-p_k-p_l \end{align} \]
\[ \begin{align} \mathcal{P}\left(k{\not\in}s{\cup}l{\not\in}s\right)&=\mathcal{P}\left(k{\notin}s\right)+\mathcal{P}\left(l{\not\in}s\right)-\mathcal{P}\left(k{\not\in}s{\cap}l{\not\in}s\right)\\ &=\mathcal{P}\left(k{\not\in}s\right)+\mathcal{P}\left(l{\not\in}s\right)-\mathcal{P}\left(k{\not\in}s{\&}l{\not\in}s\right)\\ &=\mathcal{P}\left(k{\not\in}s\right)+\mathcal{P}\left(l{\not\in}s\right)-\mathcal{P}\left(k,l{\not\in}s\right) \end{align} \]
\[ \begin{align} \pi_{kl}&=\mathcal{p}(k{\in}s{\cap}l{\in}s)\\ &=1-\mathcal{p}(k{\notin}s{\cup}l{\notin}s)\\ &=1-\left[\mathcal{p}(k{\notin}s)+\mathcal{p}(l{\notin}s)-\mathcal{p}({k,l}{\notin}s)\right]\\ &=1-\left[\left(1-p_k\right)^m+\left(1-p_l\right)^m-\binom{m}{m}{(1-p_k-p_l)}^{m}{(p_k+p_l)}^{m-m}\right]\\ &=1-{\left(1-p_k\right)}^{m}-{\left(1-p_l\right)}^{m}+{(1-p_k-p_l)}^{m} \end{align} \]
8.1.6.4 Muestra y extracción ordenada
Con \(N=5\) y \(m=2\)
- Extracciones ordenadas con reemplazo
\[ \begin{align} {N}^{m}&={5}^{2}\\ &=25 \end{align} \]
extracciones <- expand.grid('primer elemnto'=1:5,'segundo elemento'=1:5)
head(extracciones)## primer elemnto segundo elemento
## 1 1 1
## 2 2 1
## 3 3 1
## 4 4 1
## 5 5 1
## 6 1 2muestras <- OrderWR(N=5, m=2, ID=U)
head(muestras)## [,1] [,2]
## [1,] "Santiago" "Santiago"
## [2,] "Santiago" "Nestor"
## [3,] "Santiago" "Nayibe"
## [4,] "Santiago" "Raul"
## [5,] "Santiago" "Jhon"
## [6,] "Nestor" "Santiago"- Muestras con reemplazo
\[ \begin{align} \binom{N+m-1}{m}&=\binom{5+2-1}{2}\\ &=15 \end{align} \]
extracciones[extracciones$`primer elemnto`<=extracciones$`segundo elemento`,]## primer elemnto segundo elemento
## 1 1 1
## 6 1 2
## 7 2 2
## 11 1 3
## 12 2 3
## 13 3 3
## 16 1 4
## 17 2 4
## 18 3 4
## 19 4 4
## 21 1 5
## 22 2 5
## 23 3 5
## 24 4 5
## 25 5 5muestras[muestras[,1]<=muestras[,2],]## [,1] [,2]
## [1,] "Santiago" "Santiago"
## [2,] "Nestor" "Santiago"
## [3,] "Nestor" "Nestor"
## [4,] "Nestor" "Raul"
## [5,] "Nayibe" "Santiago"
## [6,] "Nayibe" "Nestor"
## [7,] "Nayibe" "Nayibe"
## [8,] "Nayibe" "Raul"
## [9,] "Raul" "Santiago"
## [10,] "Raul" "Raul"
## [11,] "Jhon" "Santiago"
## [12,] "Jhon" "Nestor"
## [13,] "Jhon" "Nayibe"
## [14,] "Jhon" "Raul"
## [15,] "Jhon" "Jhon"\[ \mathcal{p}_{k}= \begin{cases} \frac{1}{8} & \text{para }k{\in}\{Santiago, Nestor\}\\ \frac{1}{4} & \text{para }k{\in}\{Nayibe, Raul, Jhon\} \end{cases} \]
\[ \begin{align} {\sum}_U\mathcal{p}_{k}&=\mathcal{p}_{Santiago}+\mathcal{p}_{Nestor}+\mathcal{p}_{Nayibe}+\mathcal{p}_{Raul}+\mathcal{p}_{Jhon}\\ &=1 \end{align} \]
pk <- c(1/8,1/8,1/4,1/4,1/4)
sum(pk)## [1] 1muestras <- as.data.frame(OrderWR(N=5, m=2, ID=U));colnames(muestras) <- c("e1", "e2")\[p(\left\{Santiago, Santiago\right\})=\frac{2!}{2!0!0!0!0!}\left[\left(\frac{1}{8}\right)^{2}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 1,], n=t(c(2,0,0,0,0)), "p(s)"=dmultinom(x=c(2,0,0,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 1 Santiago Santiago 2 0 0 0 0 0.015625\[p(\left\{Santiago, Nestor\right\})=\frac{2!}{1!1!0!0!0!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 2,], n=t(c(1,1,0,0,0)), "p(s)"=dmultinom(x=c(1,1,0,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 2 Santiago Nestor 1 1 0 0 0 0.03125\[p(\left\{Santiago, Nayibe\right\})=\frac{2!}{1!0!1!0!0!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 3,], n=t(c(1,0,1,0,0)), "p(s)"=dmultinom(x=c(1,0,1,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 3 Santiago Nayibe 1 0 1 0 0 0.0625\[p(\left\{Santiago, Raul\right\})=\frac{2!}{1!0!0!1!0!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 4,], n=t(c(1,0,0,1,0)), "p(s)"=dmultinom(x=c(1,0,0,1,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 4 Santiago Raul 1 0 0 1 0 0.0625\[p(\left\{Santiago, Jhon\right\})=\frac{2!}{1!0!0!0!1!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[ 5,], n=t(c(1,0,0,0,1)), "p(s)"=dmultinom(x=c(1,0,0,0,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 5 Santiago Jhon 1 0 0 0 1 0.0625\[p(\left\{Nestor, Santiago\right\})=\frac{2!}{1!1!0!0!0!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 6,], n=t(c(1,1,0,0,0)), "p(s)"=dmultinom(x=c(1,1,0,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 6 Nestor Santiago 1 1 0 0 0 0.03125\[p(\left\{Nestor, Nestor\right\})=\frac{2!}{0!2!0!0!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{2}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 7,], n=t(c(0,2,0,0,0)), "p(s)"=dmultinom(x=c(0,2,0,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 7 Nestor Nestor 0 2 0 0 0 0.015625\[p(\left\{Nestor, Nayibe\right\})=\frac{2!}{0!1!1!0!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 8,], n=t(c(0,1,1,0,0)), "p(s)"=dmultinom(x=c(0,1,1,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 8 Nestor Nayibe 0 1 1 0 0 0.0625\[p(\left\{Nestor, Raul\right\})=\frac{2!}{0!1!0!1!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[ 9,], n=t(c(0,1,0,1,0)), "p(s)"=dmultinom(x=c(1,0,0,0,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 9 Nestor Raul 0 1 0 1 0 0.0625\[p(\left\{Nestor, Jhon\right\})=\frac{2!}{0!1!0!0!1!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[10,], n=t(c(0,1,0,0,1)), "p(s)"=dmultinom(x=c(0,1,0,0,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 10 Nestor Jhon 0 1 0 0 1 0.0625\[p(\left\{Nayibe, Santiago\right\})=\frac{2!}{1!0!1!0!0!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[11,], n=t(c(1,0,1,0,0)), "p(s)"=dmultinom(x=c(1,0,1,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 11 Nayibe Santiago 1 0 1 0 0 0.0625\[p(\left\{Nayibe, Nestor\right\})=\frac{2!}{0!1!1!0!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[12,], n=t(c(0,1,1,0,0)), "p(s)"=dmultinom(x=c(0,1,1,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 12 Nayibe Nestor 0 1 1 0 0 0.0625\[p(\left\{Nayibe, Nayibe\right\})=\frac{2!}{0!0!2!0!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{2}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[13,], n=t(c(0,0,2,0,0)), "p(s)"=dmultinom(x=c(0,0,2,0,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 13 Nayibe Nayibe 0 0 2 0 0 0.0625\[p(\left\{Nayibe, Raul\right\})=\frac{2!}{0!0!1!1!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[14,], n=t(c(0,0,1,1,0)), "p(s)"=dmultinom(x=c(0,0,1,1,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 14 Nayibe Raul 0 0 1 1 0 0.125\[p(\left\{Nayibe, Jhon\right\})=\frac{2!}{0!0!1!0!1!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[15,], n=t(c(0,0,1,0,1)), "p(s)"=dmultinom(x=c(0,0,1,0,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 15 Nayibe Jhon 0 0 1 0 1 0.125\[p(\left\{Raul, Santiago\right\})=\frac{2!}{1!0!0!1!0!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[16,], n=t(c(1,0,0,1,0)), "p(s)"=dmultinom(x=c(1,0,0,1,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 16 Raul Santiago 1 0 0 1 0 0.0625\[p(\left\{Raul, Nestor\right\})=\frac{2!}{0!1!0!1!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[17,], n=t(c(0,1,0,1,0)), "p(s)"=dmultinom(x=c(0,1,0,1,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 17 Raul Nestor 0 1 0 1 0 0.0625\[p(\left\{Raul, Nayibe\right\})=\frac{2!}{0!0!1!1!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[18,], n=t(c(0,0,1,1,0)), "p(s)"=dmultinom(x=c(0,0,1,1,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 18 Raul Nayibe 0 0 1 1 0 0.125\[p(\left\{Raul, Raul\right\})=\frac{2!}{0!0!0!2!0!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{2}\left(\frac{1}{4}\right)^{0}\right]\]
cbind(muestras[19,], n=t(c(0,0,0,2,0)), "p(s)"=dmultinom(x=c(0,0,0,2,0), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 19 Raul Raul 0 0 0 2 0 0.0625\[p(\left\{Raul, Jhon\right\})=\frac{2!}{0!0!0!1!1!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[20,], n=t(c(0,0,0,1,1)), "p(s)"=dmultinom(x=c(0,0,0,1,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 20 Raul Jhon 0 0 0 1 1 0.125\[p(\left\{Jhon, Santiago\right\})=\frac{2!}{1!0!0!0!1!}\left[\left(\frac{1}{8}\right)^{1}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[21,], n=t(c(1,0,0,0,1)), "p(s)"=dmultinom(x=c(1,0,0,0,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 21 Jhon Santiago 1 0 0 0 1 0.0625\[p(\left\{Jhon, Nestor\right\})=\frac{2!}{0!1!0!0!1!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[22,], n=t(c(0,1,0,0,1)), "p(s)"=dmultinom(x=c(0,1,0,0,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 22 Jhon Nestor 0 1 0 0 1 0.0625\[p(\left\{Jhon, Nayibe\right\})=\frac{2!}{0!0!1!0!1!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[23,], n=t(c(0,0,1,0,1)), "p(s)"=dmultinom(x=c(0,0,1,0,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 23 Jhon Nayibe 0 0 1 0 1 0.125\[p(\left\{Jhon, Raul\right\})=\frac{2!}{0!0!0!1!1!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{1}\left(\frac{1}{4}\right)^{1}\right]\]
cbind(muestras[24,], n=t(c(0,0,0,1,1)), "p(s)"=dmultinom(x=c(0,0,0,1,1), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 24 Jhon Raul 0 0 0 1 1 0.125\[p(\left\{Jhon, Jhon\right\})=\frac{2!}{0!0!0!0!2!}\left[\left(\frac{1}{8}\right)^{0}\left(\frac{1}{8}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{0}\left(\frac{1}{4}\right)^{2}\right]\]
cbind(muestras[25,], n=t(c(0,0,0,0,2)), "p(s)"=dmultinom(x=c(0,0,0,0,2), prob=pk))## e1 e2 n.1 n.2 n.3 n.4 n.5 p(s)
## 25 Jhon Jhon 0 0 0 0 2 0.06258.1.6.5 Estimador del total poblacional
\[U_1,U_2,\ldots,U_n\text{ variables aleatorias }i.i.d\]
\[E\left[U_i\right]=\mu\]
\[V\left[U_i\right]=\sigma^2\]
\[\overline{U}={\sum}_{i=1}^{m}\frac{U_i}{m}\]
\[ \begin{align} E\left[\overline{U}\right]&=\frac{1}{m}{\sum}_{i=1}^{m}E\left[U_i\right]\\ &=\frac{1}{m}{\sum}_{i=1}^{m}\mu\\ &=\frac{1}{m}{m}\mu\\ &=\mu \end{align} \]
\[ \begin{align} V\left[\overline{U}\right]&=\frac{1}{m^2}{\sum}_{i=1}^{m}V\left[U_i\right]\\ &=\frac{\sigma^2}{m} \end{align} \]
\[ \begin{align} {\sum}_{i=1}^{m}\left(U_i-\overline{U}\right)^2&={\sum}_{i=1}^{m}\left(U_i^2-2U_i\overline{U}+\overline{U}^2\right)\\ &={\sum}_{i=1}^{m}U_i^2-2\overline{U}{\sum}_{i=1}^{m}U_i+{\sum}_{i=1}^{m}\overline{U}^2\\ &={\sum}_{i=1}^{m}U_i^2-2\overline{U}{m}\frac{{\sum}_{i=1}^{m}U_i}{m}+{m}\overline{U}^2\\ &={\sum}_{i=1}^{m}U_i^2-2{m}\overline{U}^2+{m}\overline{U}^2\\ &={\sum}_{i=1}^{m}U_i^2-m{\overline{U}}^2 \end{align} \]
\[ \begin{align} E\left[{\sum}_{i=1}^{m}(U_i-\overline{U})^2\right]&={\sum}_{i=1}^{m}E\left[U_i^2\right]-mE\left[{\overline{U}}^2\right] \end{align} \]
\[ \begin{align} E\left[U_i^2\right]&=V\left[U_i\right]+E^2\left[U_i\right]\\ &=\sigma^2+{\mu}^2 \end{align} \]
\[ \begin{align} E\left[\overline{U}^2\right]&=V\left[\overline{U}\right]+E^2\left[\overline{U}\right]\\ &=\frac{\sigma^2}{m}+{\mu}^2 \end{align} \]
\[ \begin{align} E\left[{\sum}_{i=1}^{m}(U_i-\overline{U})^2\right]&={\sum}_{i=1}^{m}\left[\sigma^2+{\mu}^2\right]-m\left[\frac{\sigma^2}{m}+{\mu}^2\right]\\ &=m\sigma^2+m{\mu}^2-m\frac{\sigma^2}{m}-m{\mu}^2\\ &=m\sigma^2-m\frac{\sigma^2}{m}\\ &=m\sigma^2-\sigma^2\\ &=(m-1)\sigma^2 \end{align} \]
\[ \begin{align} E\left[{\sum}_{i=1}^{m}(U_i-\overline{U})^2\right]&=(m-1)\sigma^2\\ E\left[\frac{{\sum}_{i=1}^{m}(U_i-\overline{U})^2}{m-1}\right]&=\sigma^2 \end{align} \]
\[\widehat{\sigma}^2=\frac{1}{m-1}{\sum}_{i=1}^{m}(U_i-\overline{U})^2\]
\[ \begin{align} \widehat{V}\left[\overline{U}\right]&=\frac{\frac{1}{m-1}{\sum}_{i=1}^{m}(U_i-\overline{U})^2}{m}\\ &=\frac{1}{m(m-1)}{\sum}_{i=1}^{m}(U_i-\overline{U})^2 \end{align} \]
8.1.6.6 Variable aleatoria expandida
\[{\forall}_{k{\in}U,i=1,\ldots,m}Z_i=\frac{y_{k_i}}{p_{k_i}}\]
\[ \begin{align} \mathcal{P}\left(Z_i=\frac{y_{k}}{p_{k}}\right)&=\mathcal{P}\left(\frac{y_{k_i}}{p_{k_i}}=\frac{y_{k}}{p_{k}}\right)\\ &={p}_{k} \end{align} \]
\[ \begin{align} E\left(Z_i\right)&={\sum}_{U}\frac{y_{k}}{p_{k}}p_k\\ &={\sum}_{U}y_{k}\\ &={t}_{y} \end{align} \]
\[ \begin{align} V\left(Z_i\right)&={\sum}_{U}\left({\frac{y_{k}}{p_{k}}-{t}_{y}}\right)^{2}p_k \end{align} \]
\[ \begin{align} \widehat{t}_{y,p}&=\frac{1}{m}{\sum}_{i=1}^{m}\frac{y_{k_i}}{p_{k_i}} \end{align} \]
\[\forall_{k{\in}U}p_k>0{\implies}E\left(\widehat{t}_{y,p}\right)={t}_{y}\]
\[ \begin{align} E\left[\widehat{t}_{y,p}\right]&=\frac{1}{m}{\sum}_{i=1}^{m}E\left[{Z_{i}}\right]\\ &=\frac{1}{m}{\sum}_{i=1}^{m}{\sum}_{k=1}^{N}\frac{y_{k}}{p_{k}}p_k\\ &=\frac{1}{m}{m}{\sum}_{k=1}^{N}\frac{y_{k}}{p_{k}}p_k\\ &={\sum}_{k=1}^{N}\frac{y_{k}}{p_{k}}p_k\\ &={\sum}_{k=1}^{N}{y_{k}}\\ &={t}_{y} \end{align} \]
\[ \begin{align} V\left[\widehat{t}_{y,p}\right]&=\frac{1}{{m}^{2}}{\sum}_{i=1}^{m}V\left[{Z_{i}}\right]\\ &=\frac{1}{{m}^{2}}{\sum}_{i=1}^{m}E\left[{Z_{i}}-E\left({Z_{i}}\right)\right]^{2}\\ &=\frac{1}{{m}^{2}}{m}E\left[{Z_{k}}-{t}_{y}\right]^{2}\\ &=\frac{1}{{m}}E\left[{Z_{k}}-{t}_{y}\right]^{2}\\ &=\frac{1}{{m}}{\sum}_{i=1}^{N}\left[{Z_{k}}-{t}_{y}\right]^{2}p_k\\ &=\frac{1}{{m}}{\sum}_{i=1}^{N}\left[\frac{y_{k}}{p_{k}}-{t}_{y}\right]^{2}p_k \end{align} \]
\[ \begin{align} V\left[\widehat{t}_{y,p}\right]&=\frac{1}{m}{\sum}_{i=1}^{N}\left[\frac{y_{k}}{p_{k}}-{t}_{y}\right]^{2}p_k\\ &=\frac{1}{m}{\sum}_{i=1}^{N}\left[\frac{y_{k}^{2}}{p_{k}^{2}}-2\frac{y_{k}}{p_{k}}{t}_{y}+{t}_{y}^{2}\right]p_k\\ &=\frac{1}{m}{\sum}_{i=1}^{N}\left[p_k\frac{y_{k}^{2}}{p_{k}^{2}}-2p_k\frac{y_{k}}{p_{k}}{t}_{y}+p_k{t}_{y}^{2}\right]\\ &=\frac{1}{m}{\sum}_{i=1}^{N}\left[\frac{y_{k}^{2}}{p_{k}}-2{y_{k}}{t}_{y}+p_k{t}_{y}^{2}\right]\\ &=\frac{1}{m}\left[{\sum}_{i=1}^{N}\frac{y_{k}^{2}}{p_{k}}-2{t}_{y}{\sum}_{i=1}^{N}{y_{k}}+{t}_{y}^{2}{\sum}_{i=1}^{N}p_k\right]\\ &=\frac{1}{m}\left[{\sum}_{i=1}^{N}\frac{y_{k}^{2}}{p_{k}}-2{t}_{y}{t}_{y}+{t}_{y}^{2}\right]\\ &=\frac{1}{m}\left[{\sum}_{i=1}^{N}\frac{y_{k}^{2}}{p_{k}}-{t}_{y}^{2}\right] \end{align} \]
\[ \begin{align} \widehat{V}\left[\widehat{t}_{y,p}\right]&=\frac{1}{m\left(m-1\right)}{\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}-\widehat{t}_{y,p}\right]^{2}\\ \end{align} \]
\[ \begin{align} \frac{1}{m}E\left[{Z_{k}}-{t}_{y}\right]^{2}&=\frac{1}{m}V\left[{Z_{i}}\right]\\ \frac{1}{m}\widehat{V}\left[{Z_{i}}\right]&=\frac{1}{m}\frac{1}{m-1}{\sum}_{i=1}^{m}\left[{Z_{i}}-\bar{Z}\right]^{2}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}-\widehat{t}_{y,p}\right]^{2} \end{align} \]
\[ \begin{align} m\left(m-1\right)\widehat{V}\left[\widehat{t}_{y,p}\right]&={\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}-\widehat{t}_{y,p}\right]^{2}\\ &={\sum}_{i=1}^{m}\left[\frac{y_{i}^{2}}{p_{i}^{2}}-2\widehat{t}_{y}\frac{y_{i}}{p_{i}}+\widehat{t}_{y,p}^{2}\right]\\ &={\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}\right]^{2}-2\widehat{t}_{y,p}{\sum}_{i=1}^{m}\frac{y_{i}}{p_{i}}+m\widehat{t}_{y,p}^{2}\\ &={\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}\right]^{2}-2\widehat{t}_{y,p}\frac{m}{m}{\sum}_{i=1}^{m}\frac{y_{i}}{p_{i}}+m\widehat{t}_{y,p}^{2}\\ &={\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}\right]^{2}-2\widehat{t}_{y,p}{m}\frac{1}{m}{\sum}_{i=1}^{m}\frac{y_{i}}{p_{i}}+m\widehat{t}_{y,p}^{2}\\ &={\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}\right]^{2}-2\widehat{t}_{y,p}{m}\widehat{t}_{y,p}+{m}\widehat{t}_{y,p}^{2}\\ &={\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}\right]^{2}-2{m}\widehat{t}_{y,p}^{2}+m\widehat{t}_{y,p}^{2}\\ &={\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}\right]^{2}-{m}\widehat{t}_{y,p}^{2}\\ \widehat{V}\left[\widehat{t}_{y,p}\right]&=\frac{1}{m\left(m-1\right)}\left\{{\sum}_{i=1}^{m}\left[\frac{y_{i}}{p_{i}}\right]^{2}-{m}\widehat{t}_{y,p}^{2}\right\} \end{align} \]
y## yk
## Santiago 32
## Nestor 442
## Nayibe 5454
## Raul 646
## Jhon 656pk <- c(.5, .12, .20, .155, .025)
sum(pk)## [1] 1s <- sample(x=5, size=3, replace=TRUE, prob=pk)
s## [1] 4 4 3pkm <- pk[s]
pkm## [1] 0.155 0.155 0.200ym <- y[s]
ym## [1] 646 646 5454typ <- mean(ym[1:3]/pkm[1:3])
typ## [1] 11868.49typ <- HH(ym,pkm)
typ## y
## Estimation 11868.49462
## Standard Error 7700.75269
## CVE 64.88399Vtyp <- (1/3)*(1/(3-1))*sum((ym[1:3]/pkm[1:3]-typ["Estimation","y"])**2)
Vtyp## [1] 59301592cve <- sqrt(Vtyp)/typ["Estimation","y"]
cve## [1] 0.64883998.2 Estrategia \(\left(\widehat{t}_{y,\pi},p(s)\right)\)
8.2.1 \(\widehat{t}_{y,\pi}={\sum}_{s}\frac{y_k}{\pi_k}\)
\[ \begin{align} \widehat{t}_{y,\pi}&={\sum}_{s}d_ky_k \end{align} \]
data.frame(Q,p,Ind,typi)## X1 X2 p X1.1 X2.1 X3 X4 X5 ybars
## 1 Santiago Nestor 0.13 1 1 0 0 0 1355.172
## 2 Santiago Nayibe 0.20 1 0 1 0 0 11417.672
## 3 Santiago Raul 0.15 1 0 0 1 0 2012.748
## 4 Santiago Jhon 0.10 1 0 0 0 1 2484.802
## 5 Nestor Nayibe 0.15 0 1 1 0 0 12662.500
## 6 Nestor Raul 0.04 0 1 0 1 0 3257.576
## 7 Nestor Jhon 0.02 0 1 0 0 1 3729.630
## 8 Nayibe Raul 0.06 0 0 1 1 0 13320.076
## 9 Nayibe Jhon 0.07 0 0 1 0 1 13792.130
## 10 Raul Jhon 0.08 0 0 0 1 1 4387.2058.2.1.1 \(s=\left\{Santiago,Nestor\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Santiago}+\breve{y}_{Nestor}\\ &=\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Nestor}}{\pi_{Nestor}} \end{align} \]
typi <- y[1]/pik[1]+y[2]/pik[2]
typi## [1] 1355.172\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Santiago},{Nestor}\}}\frac{y_k}{\pi_k}\]
HT(y[c(1,2)],pik[c(1,2)])## [,1]
## [1,] 1355.1728.2.1.2 \(s=\left\{Santiago,Nayibe\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Santiago}+\breve{y}_{Nayibe}\\ &=\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Nayibe}}{\pi_{Nayibe}} \end{align} \]
typi <- y[1]/pik[1]+y[3]/pik[3]
typi## [1] 11417.67\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Santiago},{Nayibe}\}}\frac{y_k}{\pi_k}\]
HT(y[c(1,3)],pik[c(1,3)])## [,1]
## [1,] 11417.678.2.1.3 \(s=\left\{Santiago,Raul\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Santiago}+\breve{y}_{Raul}\\ &=\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Raul}}{\pi_{Raul}} \end{align} \]
typi <- y[1]/pik[1]+y[4]/pik[4]
typi## [1] 2012.748\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Santiago},{Raul}\}}\frac{y_k}{\pi_k}\]
HT(y[c(1,4)],pik[c(1,4)])## [,1]
## [1,] 2012.7488.2.1.4 \(s=\left\{Santiago,Jhon\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Santiago}+\breve{y}_{Jhon}\\ &=\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Jhon}}{\pi_{Jhon}} \end{align} \]
typi <- y[1]/pik[1]+y[5]/pik[5]
typi## [1] 2484.802\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Santiago},{Jhon}\}}\frac{y_k}{\pi_k}\]
HT(y[c(1,5)],pik[c(1,5)])## [,1]
## [1,] 2484.8028.2.1.5 \(s=\left\{Nestor,Nayibe\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Nestor}+\breve{y}_{Nayibe}\\ &=\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{y_{Nayibe}}{\pi_{Nayibe}} \end{align} \]
typi <- y[2]/pik[2]+y[3]/pik[3]
typi## [1] 12662.5\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Nestor},{Nayibe}\}}\frac{y_k}{\pi_k}\]
HT(y[c(2,3)],pik[c(2,3)])## [,1]
## [1,] 12662.58.2.1.6 \(s=\left\{Nestor,Raul\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Nestor}+\breve{y}_{Raul}\\ &=\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{y_{Raul}}{\pi_{Raul}} \end{align} \]
typi <- y[2]/pik[2]+y[4]/pik[4]
typi## [1] 3257.576\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Nestor},{Raul}\}}\frac{y_k}{\pi_k}\]
HT(y[c(2,4)],pik[c(2,4)])## [,1]
## [1,] 3257.5768.2.1.7 \(s=\left\{Nestor,Jhon\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Nestor}+\breve{y}_{Jhon}\\ &=\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{y_{Jhon}}{\pi_{Jhon}} \end{align} \]
typi <- y[2]/pik[2]+y[5]/pik[5]
typi## [1] 3729.63\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Nestor},{Jhon}\}}\frac{y_k}{\pi_k}\]
HT(y[c(2,5)],pik[c(2,5)])## [,1]
## [1,] 3729.638.2.1.8 \(s=\left\{Nayibe,Raul\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Nayibe}+\breve{y}_{Raul}\\ &=\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{y_{Raul}}{\pi_{Raul}} \end{align} \]
typi <- y[3]/pik[3]+y[4]/pik[4]
typi## [1] 13320.08\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Nayibe},{Raul}\}}\frac{y_k}{\pi_k}\]
HT(y[c(3,4)],pik[c(3,4)])## [,1]
## [1,] 13320.088.2.1.9 \(s=\left\{Nayibe,Jhon\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Nayibe}+\breve{y}_{Jhon}\\ &=\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{y_{Jhon}}{\pi_{Jhon}} \end{align} \]
typi <- y[3]/pik[3]+y[5]/pik[5]
typi## [1] 13792.13\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Nayibe},{Jhon}\}}\frac{y_k}{\pi_k}\]
HT(y[c(3,5)],pik[c(3,5)])## [,1]
## [1,] 13792.138.2.1.10 \(s=\left\{Raul,Jhon\right\}\)
\[ \begin{align} \widehat{t}_{y,\pi}&=\breve{y}_{Raul}+\breve{y}_{Jhon}\\ &=\frac{y_{Raul}}{\pi_{Raul}}+\frac{y_{Jhon}}{\pi_{Jhon}} \end{align} \]
typi <- y[4]/pik[4]+y[5]/pik[5]
typi## [1] 4387.205\[\widehat{t}_{y,\pi}={\sum}_{k{\in}\{{Raul},{Jhon}\}}\frac{y_k}{\pi_k}\]
HT(y[c(4,5)],pik[c(4,5)])## [,1]
## [1,] 4387.2058.2.2 \(\widehat{V}[\widehat{t}_{y,\pi}]={{\sum}{\sum}}_{s}\frac{\Delta_{kl}}{\pi_{kl}}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l}\)
8.2.2.1 \(s=\left\{Santiago,Nestor\right\}\)
\[ \Delta_{{Santiago},{Santiago}}=\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}} \]
delta11 <- pikl["Santiago","Santiago"]-pikl["Santiago","Santiago"]*pikl["Santiago","Santiago"]
delta11## [1] 0.2436\[ \Delta_{{Santiago},{Nestor}}=\pi_{{Santiago},{Nestor}}-\pi_{{Santiago}}\pi_{{Nestor}} \]
delta12 <- pikl["Santiago","Nestor"]-pikl["Santiago","Santiago"]*pikl["Nestor","Nestor"]
delta12## [1] -0.0672\[ \Delta_{{Nestor},{Santiago}}=\pi_{{Nestor},{Santiago}}-\pi_{{Nestor}}\pi_{{Santiago}} \]
delta21 <- pikl["Nestor","Santiago"]-pikl["Nestor","Nestor"]*pikl["Santiago","Santiago"]
delta21## [1] -0.0672\[ \Delta_{{Nestor},{Nestor}}=\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}} \]
delta22 <- pikl["Nestor","Nestor"]-pikl["Nestor","Nestor"]*pikl["Nestor","Nestor"]
delta22## [1] 0.2244\[ \frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}=\frac{\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}}}{\pi_{{Santiago},{Santiago}}} \]
delta11/pikl["Santiago","Santiago"]## [1] 0.42\[ \frac{\Delta_{{Santiago},{Nestor}}}{\pi_{{Santiago},{Nestor}}}=\frac{\pi_{{Santiago},{Nestor}}-\pi_{{Santiago}}\pi_{{Nestor}}}{\pi_{{Santiago},{Nestor}}} \]
delta12/pikl["Santiago","Nestor"]## [1] -0.5169231\[ \frac{\Delta_{{Nestor},{Santiago}}}{\pi_{{Nestor},{Santiago}}}=\frac{\pi_{{Nestor},{Santiago}}-\pi_{{Nestor}}\pi_{{Santiago}}}{\pi_{{Nestor},{Santiago}}} \]
delta21/pikl["Nestor","Santiago"]## [1] -0.5169231\[ \frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}=\frac{\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}}}{\pi_{{Nestor},{Nestor}}} \]
delta22/pikl["Nestor","Nestor"]## [1] 0.66\[y_{Santiago}\]
y1 <- y[1]
y1## [1] 32\[y_{Nestor}\]
y2 <- y[2]
y2## [1] 442\[\breve{y}_{Santiago}=\frac{y_{Santiago}}{\pi_{Santiago}}\]
y1/pik[1]## [1] 55.17241\[\breve{y}_{Nestor}=\frac{y_{Nestor}}{\pi_{Nestor}}\]
y2/pik[2]## [1] 1300\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Nestor}}}{\pi_{{Santiago},{Nestor}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}} \end{align} \]
Vtypi <- delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta12/pikl["Santiago","Nestor"]*(y1/pik[1])*(y2/pik[2])+delta22/pikl["Nestor","Nestor"]*(y2/pik[2])**2
Vtypi## [1] 10425278.2.2.2 \(s=\left\{Santiago,Nayibe\right\}\)
\[ \Delta_{{Santiago},{Santiago}}=\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}} \]
delta11 <- pikl["Santiago","Santiago"]-pikl["Santiago","Santiago"]*pikl["Santiago","Santiago"]
delta11## [1] 0.2436\[ \Delta_{{Santiago},{Nayibe}}=\pi_{{Santiago},{Nayibe}}-\pi_{{Santiago}}\pi_{{Nayibe}} \]
delta13 <- pikl["Santiago","Nayibe"]-pikl["Santiago","Santiago"]*pikl["Nayibe","Nayibe"]
delta13## [1] -0.0784\[ \Delta_{{Nayibe},{Santiago}}=\pi_{{Nayibe},{Santiago}}-\pi_{{Nayibe}}\pi_{{Santiago}} \]
delta31 <- pikl["Nayibe","Santiago"]-pikl["Nayibe","Nayibe"]*pikl["Santiago","Santiago"]
delta31## [1] -0.0784\[ \Delta_{{Nayibe},{Nayibe}}=\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}} \]
delta33 <- pikl["Nayibe","Nayibe"]-pikl["Nayibe","Nayibe"]*pikl["Nayibe","Nayibe"]
delta33## [1] 0.2496\[ \frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}=\frac{\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}}}{\pi_{{Santiago},{Santiago}}} \]
delta11/pikl["Santiago","Santiago"]## [1] 0.42\[ \frac{\Delta_{{Santiago},{Nayibe}}}{\pi_{{Santiago},{Nayibe}}}=\frac{\pi_{{Santiago},{Nayibe}}-\pi_{{Santiago}}\pi_{{Nayibe}}}{\pi_{{Santiago},{Nayibe}}} \]
delta13/pikl["Santiago","Nayibe"]## [1] -0.392\[ \frac{\Delta_{{Nayibe},{Santiago}}}{\pi_{{Nayibe},{Santiago}}}=\frac{\pi_{{Nayibe},{Santiago}}-\pi_{{Nayibe}}\pi_{{Santiago}}}{\pi_{{Nayibe},{Santiago}}} \]
delta31/pikl["Nayibe","Santiago"]## [1] -0.392\[ \frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}=\frac{\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}} \]
delta33/pikl["Nayibe","Nayibe"]## [1] 0.52\[y_{Santiago}\]
y1 <- y[1]
y1## [1] 32\[y_{Nayibe}\]
y3 <- y[3]
y3## [1] 5454\[\breve{y}_{Santiago}=\frac{y_{Santiago}}{\pi_{Santiago}}\]
y1/pik[1]## [1] 55.17241\[\breve{y}_{Nayibe}=\frac{y_{Nayibe}}{\pi_{Nayibe}}\]
y3/pik[3]## [1] 11362.5\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Nayibe}}}{\pi_{{Santiago},{Nayibe}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}} \end{align} \]
Vtypi <- delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta13/pikl["Santiago","Nayibe"]*(y1/pik[1])*(y3/pik[3])+delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2
Vtypi## [1] 666451238.2.2.3 \(s=\left\{Santiago,Raul\right\}\)
\[ \Delta_{{Santiago},{Santiago}}=\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}} \]
delta11 <- pikl["Santiago","Santiago"]-pikl["Santiago","Santiago"]*pikl["Santiago","Santiago"]
delta11## [1] 0.2436\[ \Delta_{{Santiago},{Raul}}=\pi_{{Santiago},{Raul}}-\pi_{{Santiago}}\pi_{{Raul}} \]
delta14 <- pikl["Santiago","Raul"]-pikl["Santiago","Santiago"]*pikl["Raul","Raul"]
delta14## [1] -0.0414\[ \Delta_{{Raul},{Santiago}}=\pi_{{Raul},{Santiago}}-\pi_{{Raul}}\pi_{{Santiago}} \]
delta41 <- pikl["Raul","Santiago"]-pikl["Raul","Raul"]*pikl["Santiago","Santiago"]
delta41## [1] -0.0414\[ \Delta_{{Raul},{Raul}}=\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}} \]
delta44 <- pikl["Raul","Raul"]-pikl["Raul","Raul"]*pikl["Raul","Raul"]
delta44## [1] 0.2211\[ \frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}=\frac{\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}}}{\pi_{{Santiago},{Santiago}}} \]
delta11/pikl["Santiago","Santiago"]## [1] 0.42\[ \frac{\Delta_{{Santiago},{Raul}}}{\pi_{{Santiago},{Raul}}}=\frac{\pi_{{Santiago},{Raul}}-\pi_{{Santiago}}\pi_{{Raul}}}{\pi_{{Santiago},{Raul}}} \]
delta14/pikl["Santiago","Raul"]## [1] -0.276\[ \frac{\Delta_{{Raul},{Santiago}}}{\pi_{{Raul},{Santiago}}}=\frac{\pi_{{Raul},{Santiago}}-\pi_{{Raul}}\pi_{{Santiago}}}{\pi_{{Raul},{Santiago}}} \]
delta41/pikl["Raul","Santiago"]## [1] -0.276\[ \frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}=\frac{\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}}}{\pi_{{Raul},{Raul}}} \]
delta44/pikl["Raul","Raul"]## [1] 0.67\[y_{Santiago}\]
y1 <- y[1]
y1## [1] 32\[y_{Raul}\]
y4 <- y[4]
y4## [1] 646\[\breve{y}_{Santiago}=\frac{y_{Santiago}}{\pi_{Santiago}}\]
y1/pik[1]## [1] 55.17241\[\breve{y}_{Raul}=\frac{y_{Raul}}{\pi_{Raul}}\]
y4/pik[4]## [1] 1957.576\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Raul}}}{\pi_{{Santiago},{Raul}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Raul}}{\pi_{Raul}}+\frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}\frac{y_{Raul}^{2}}{\pi_{Raul}^{2}} \end{align} \]
Vtypi <- delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta14/pikl["Santiago","Raul"]*(y1/pik[1])*(y4/pik[4])+delta44/pikl["Raul","Raul"]*(y4/pik[4])**2
Vtypi## [1] 25091698.2.2.4 \(s=\left\{Santiago,Jhon\right\}\)
\[ \Delta_{{Santiago},{Santiago}}=\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}} \]
delta11 <- pikl["Santiago","Santiago"]-pikl["Santiago","Santiago"]*pikl["Santiago","Santiago"]
delta11## [1] 0.2436\[ \Delta_{{Santiago},{Jhon}}=\pi_{{Santiago},{Jhon}}-\pi_{{Santiago}}\pi_{{Jhon}} \]
delta15 <- pikl["Santiago","Jhon"]-pikl["Santiago","Santiago"]*pikl["Jhon","Jhon"]
delta15## [1] -0.0566\[ \Delta_{{Jhon},{Santiago}}=\pi_{{Jhon},{Santiago}}-\pi_{{Jhon}}\pi_{{Santiago}} \]
delta51 <- pikl["Jhon","Santiago"]-pikl["Jhon","Jhon"]*pikl["Santiago","Santiago"]
delta51## [1] -0.0566\[ \Delta_{{Jhon},{Jhon}}=\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}} \]
delta55 <- pikl["Jhon","Jhon"]-pikl["Jhon","Jhon"]*pikl["Jhon","Jhon"]
delta55## [1] 0.1971\[ \frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}=\frac{\pi_{{Santiago},{Santiago}}-\pi_{{Santiago}}\pi_{{Santiago}}}{\pi_{{Santiago},{Santiago}}} \]
delta11/pikl["Santiago","Santiago"]## [1] 0.42\[ \frac{\Delta_{{Santiago},{Jhon}}}{\pi_{{Santiago},{Jhon}}}=\frac{\pi_{{Santiago},{Jhon}}-\pi_{{Santiago}}\pi_{{Jhon}}}{\pi_{{Santiago},{Jhon}}} \]
delta15/pikl["Santiago","Jhon"]## [1] -0.566\[ \frac{\Delta_{{Jhon},{Santiago}}}{\pi_{{Jhon},{Santiago}}}=\frac{\pi_{{Jhon},{Santiago}}-\pi_{{Jhon}}\pi_{{Santiago}}}{\pi_{{Jhon},{Santiago}}} \]
delta51/pikl["Jhon","Santiago"]## [1] -0.566\[ \frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}=\frac{\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}}}{\pi_{{Jhon},{Jhon}}} \]
delta55/pikl["Jhon","Jhon"]## [1] 0.73\[y_{Santiago}\]
y1 <- y[1]
y1## [1] 32\[y_{Jhon}\]
y5 <- y[5]
y5## [1] 656\[\breve{y}_{Santiago}=\frac{y_{Santiago}}{\pi_{Santiago}}\]
y1/pik[1]## [1] 55.17241\[\breve{y}_{Jhon}=\frac{y_{Jhon}}{\pi_{Jhon}}\]
y5/pik[5]## [1] 2429.63\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Jhon}}}{\pi_{{Santiago},{Jhon}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}} \end{align} \]
Vtypi <- delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta15/pikl["Santiago","Jhon"]*(y1/pik[1])*(y5/pik[5])+delta55/pikl["Jhon","Jhon"]*(y5/pik[5])**2
Vtypi## [1] 41587998.2.2.5 \(s=\left\{Nestor,Nayibe\right\}\)
\[ \Delta_{{Nestor},{Nestor}}=\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}} \]
delta22 <- pikl["Nestor","Nestor"]-pikl["Nestor","Nestor"]*pikl["Nestor","Nestor"]
delta22## [1] 0.2244\[ \Delta_{{Nestor},{Nayibe}}=\pi_{{Nestor},{Nayibe}}-\pi_{{Nestor}}\pi_{{Nayibe}} \]
delta23 <- pikl["Nestor","Nayibe"]-pikl["Nestor","Nestor"]*pikl["Nayibe","Nayibe"]
delta23## [1] -0.0132\[ \Delta_{{Nayibe},{Nestor}}=\pi_{{Nayibe},{Nestor}}-\pi_{{Nayibe}}\pi_{{Nestor}} \]
delta32 <- pikl["Nayibe","Nestor"]-pikl["Nayibe","Nayibe"]*pikl["Nestor","Nestor"]
delta32## [1] -0.0132\[ \Delta_{{Nayibe},{Nayibe}}=\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}} \]
delta33 <- pikl["Nayibe","Nayibe"]-pikl["Nayibe","Nayibe"]*pikl["Nayibe","Nayibe"]
delta33## [1] 0.2496\[ \frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}=\frac{\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}}}{\pi_{{Nestor},{Nestor}}} \]
delta22/pikl["Nestor","Nestor"]## [1] 0.66\[ \frac{\Delta_{{Nestor},{Nayibe}}}{\pi_{{Nestor},{Nayibe}}}=\frac{\pi_{{Nestor},{Nayibe}}-\pi_{{Nestor}}\pi_{{Nayibe}}}{\pi_{{Nestor},{Nayibe}}} \]
delta23/pikl["Nestor","Nayibe"]## [1] -0.088\[ \frac{\Delta_{{Nayibe},{Nestor}}}{\pi_{{Nayibe},{Nestor}}}=\frac{\pi_{{Nayibe},{Nestor}}-\pi_{{Nayibe}}\pi_{{Nestor}}}{\pi_{{Nayibe},{Nestor}}} \]
delta32/pikl["Nayibe","Nestor"]## [1] -0.088\[ \frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}=\frac{\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}} \]
delta33/pikl["Nayibe","Nayibe"]## [1] 0.52\[y_{Nestor}\]
y2 <- y[2]
y2## [1] 442\[y_{Nayibe}\]
y3 <- y[3]
y3## [1] 5454\[\breve{y}_{Nestor}=\frac{y_{Nestor}}{\pi_{Nestor}}\]
y2/pik[2]## [1] 1300\[\breve{y}_{Nayibe}=\frac{y_{Nayibe}}{\pi_{Nayibe}}\]
y3/pik[3]## [1] 11362.5\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}+2\frac{\Delta_{{Nestor},{Nayibe}}}{\pi_{{Nestor},{Nayibe}}}\frac{y_{Nestor}}{\pi_{Nestor}}\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}} \end{align} \]
Vtypi <- delta22/pikl["Nestor","Nestor"]*(y2/pik[2])**2+2*delta23/pikl["Nestor","Nayibe"]*(y2/pik[2])*(y3/pik[3])+delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2
Vtypi## [1] 656509918.2.2.6 \(s=\left\{Nestor,Raul\right\}\)
\[ \Delta_{{Nestor},{Nestor}}=\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}} \]
delta22 <- pikl["Nestor","Nestor"]-pikl["Nestor","Nestor"]*pikl["Nestor","Nestor"]
delta22## [1] 0.2244\[ \Delta_{{Nestor},{Raul}}=\pi_{{Nestor},{Raul}}-\pi_{{Nestor}}\pi_{{Raul}} \]
delta24 <- pikl["Nestor","Raul"]-pikl["Nestor","Nestor"]*pikl["Raul","Raul"]
delta24## [1] -0.0722\[ \Delta_{{Raul},{Nestor}}=\pi_{{Raul},{Nestor}}-\pi_{{Raul}}\pi_{{Nestor}} \]
delta42 <- pikl["Raul","Nestor"]-pikl["Raul","Raul"]*pikl["Nestor","Nestor"]
delta42## [1] -0.0722\[ \Delta_{{Raul},{Raul}}=\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}} \]
delta44 <- pikl["Raul","Raul"]-pikl["Raul","Raul"]*pikl["Raul","Raul"]
delta44## [1] 0.2211\[ \frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}=\frac{\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}}}{\pi_{{Nestor},{Nestor}}} \]
delta22/pikl["Nestor","Nestor"]## [1] 0.66\[ \frac{\Delta_{{Nestor},{Raul}}}{\pi_{{Nestor},{Raul}}}=\frac{\pi_{{Nestor},{Raul}}-\pi_{{Nestor}}\pi_{{Raul}}}{\pi_{{Nestor},{Raul}}} \]
delta24/pikl["Nestor","Raul"]## [1] -1.805\[ \frac{\Delta_{{Raul},{Nestor}}}{\pi_{{Raul},{Nestor}}}=\frac{\pi_{{Raul},{Nestor}}-\pi_{{Raul}}\pi_{{Nestor}}}{\pi_{{Raul},{Nestor}}} \]
delta42/pikl["Raul","Nestor"]## [1] -1.805\[ \frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}=\frac{\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}}}{\pi_{{Raul},{Raul}}} \]
delta44/pikl["Raul","Raul"]## [1] 0.67\[y_{Nestor}\]
y2 <- y[2]
y2## [1] 442\[y_{Raul}\]
y4 <- y[4]
y4## [1] 646\[\breve{y}_{Nestor}=\frac{y_{Nestor}}{\pi_{Nestor}}\]
y2/pik[2]## [1] 1300\[\breve{y}_{Raul}=\frac{y_{Raul}}{\pi_{Raul}}\]
y4/pik[4]## [1] 1957.576\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}+2\frac{\Delta_{{Nestor},{Raul}}}{\pi_{{Nestor},{Raul}}}\frac{y_{Nestor}}{\pi_{Nestor}}\frac{y_{Raul}}{\pi_{Raul}}+\frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}\frac{y_{Raul}^{2}}{\pi_{Raul}^{2}} \end{align} \]
Vtypi <- delta22/pikl["Nestor","Nestor"]*(y2/pik[2])**2+2*delta24/pikl["Nestor","Raul"]*(y2/pik[2])*(y4/pik[4])+delta44/pikl["Raul","Raul"]*(y4/pik[4])**2
Vtypi## [1] -55039948.2.2.7 \(s=\left\{Nestor,Jhon\right\}\)
\[ \Delta_{{Nestor},{Nestor}}=\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}} \]
delta22 <- pikl["Nestor","Nestor"]-pikl["Nestor","Nestor"]*pikl["Nestor","Nestor"]
delta22## [1] 0.2244\[ \Delta_{{Nestor},{Jhon}}=\pi_{{Nestor},{Jhon}}-\pi_{{Nestor}}\pi_{{Jhon}} \]
delta25 <- pikl["Nestor","Jhon"]-pikl["Nestor","Nestor"]*pikl["Jhon","Jhon"]
delta25## [1] -0.0718\[ \Delta_{{Jhon},{Nestor}}=\pi_{{Jhon},{Nestor}}-\pi_{{Jhon}}\pi_{{Nestor}} \]
delta52 <- pikl["Jhon","Nestor"]-pikl["Jhon","Jhon"]*pikl["Nestor","Nestor"]
delta52## [1] -0.0718\[ \Delta_{{Jhon},{Jhon}}=\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}} \]
delta55 <- pikl["Jhon","Jhon"]-pikl["Jhon","Jhon"]*pikl["Jhon","Jhon"]
delta55## [1] 0.1971\[ \frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}=\frac{\pi_{{Nestor},{Nestor}}-\pi_{{Nestor}}\pi_{{Nestor}}}{\pi_{{Nestor},{Nestor}}} \]
delta22/pikl["Nestor","Nestor"]## [1] 0.66\[ \frac{\Delta_{{Nestor},{Jhon}}}{\pi_{{Nestor},{Jhon}}}=\frac{\pi_{{Nestor},{Jhon}}-\pi_{{Nestor}}\pi_{{Jhon}}}{\pi_{{Nestor},{Jhon}}} \]
delta25/pikl["Nestor","Jhon"]## [1] -3.59\[ \frac{\Delta_{{Jhon},{Nestor}}}{\pi_{{Jhon},{Nestor}}}=\frac{\pi_{{Jhon},{Nestor}}-\pi_{{Jhon}}\pi_{{Nestor}}}{\pi_{{Jhon},{Nestor}}} \]
delta52/pikl["Jhon","Nestor"]## [1] -3.59\[ \frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}=\frac{\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}}}{\pi_{{Jhon},{Jhon}}} \]
delta55/pikl["Jhon","Jhon"]## [1] 0.73\[y_{Nestor}\]
y2 <- y[2]
y2## [1] 442\[y_{Jhon}\]
y5 <- y[5]
y5## [1] 656\[\breve{y}_{Nestor}=\frac{y_{Nestor}}{\pi_{Nestor}}\]
y2/pik[2]## [1] 1300\[\breve{y}_{Jhon}=\frac{y_{Jhon}}{\pi_{Jhon}}\]
y5/pik[5]## [1] 2429.63\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}+2\frac{\Delta_{{Nestor},{Jhon}}}{\pi_{{Nestor},{Jhon}}}\frac{y_{Nestor}}{\pi_{Nestor}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}} \end{align} \]
Vtypi <- delta22/pikl["Nestor","Nestor"]*(y2/pik[2])**2+2*delta25/pikl["Nestor","Jhon"]*(y2/pik[2])*(y5/pik[5])+delta55/pikl["Jhon","Jhon"]*(y5/pik[5])**2
Vtypi## [1] -172535008.2.2.8 \(s=\left\{Nayibe,Raul\right\}\)
\[ \Delta_{{Nayibe},{Nayibe}}=\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}} \]
delta33 <- pikl["Nayibe","Nayibe"]-pikl["Nayibe","Nayibe"]*pikl["Nayibe","Nayibe"]
delta33## [1] 0.2496\[ \Delta_{{Nayibe},{Raul}}=\pi_{{Nayibe},{Raul}}-\pi_{{Nayibe}}\pi_{{Raul}} \]
delta34 <- pikl["Nayibe","Raul"]-pikl["Nayibe","Nayibe"]*pikl["Raul","Raul"]
delta34## [1] -0.0984\[ \Delta_{{Raul},{Nayibe}}=\pi_{{Raul},{Nayibe}}-\pi_{{Raul}}\pi_{{Nayibe}} \]
delta43 <- pikl["Raul","Nayibe"]-pikl["Raul","Raul"]*pikl["Nayibe","Nayibe"]
delta43## [1] -0.0984\[ \Delta_{{Raul},{Raul}}=\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}} \]
delta44 <- pikl["Raul","Raul"]-pikl["Raul","Raul"]*pikl["Raul","Raul"]
delta44## [1] 0.2211\[ \frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}=\frac{\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}} \]
delta33/pikl["Nayibe","Nayibe"]## [1] 0.52\[ \frac{\Delta_{{Nayibe},{Raul}}}{\pi_{{Nayibe},{Raul}}}=\frac{\pi_{{Nayibe},{Raul}}-\pi_{{Nayibe}}\pi_{{Raul}}}{\pi_{{Nayibe},{Raul}}} \]
delta34/pikl["Nayibe","Raul"]## [1] -1.64\[ \frac{\Delta_{{Raul},{Nayibe}}}{\pi_{{Raul},{Nayibe}}}=\frac{\pi_{{Raul},{Nayibe}}-\pi_{{Raul}}\pi_{{Nayibe}}}{\pi_{{Raul},{Nayibe}}} \]
delta43/pikl["Raul","Nayibe"]## [1] -1.64\[ \frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}=\frac{\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}}}{\pi_{{Raul},{Raul}}} \]
delta44/pikl["Raul","Raul"]## [1] 0.67\[y_{Nayibe}\]
y3 <- y[3]
y3## [1] 5454\[y_{Raul}\]
y4 <- y[4]
y4## [1] 646\[\breve{y}_{Nayibe}=\frac{y_{Nayibe}}{\pi_{Nayibe}}\]
y3/pik[3]## [1] 11362.5\[\breve{y}_{Raul}=\frac{y_{Raul}}{\pi_{Raul}}\]
y4/pik[4]## [1] 1957.576\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}}+2\frac{\Delta_{{Nayibe},{Raul}}}{\pi_{{Nayibe},{Raul}}}\frac{y_{Nayibe}}{\pi_{Nayibe}}\frac{y_{Raul}}{\pi_{Raul}}+\frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}\frac{y_{Raul}^{2}}{\pi_{Raul}^{2}} \end{align} \]
Vtypi <- delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2+2*delta34/pikl["Nayibe","Raul"]*(y3/pik[3])*(y4/pik[4])+delta44/pikl["Raul","Raul"]*(y4/pik[4])**2
Vtypi## [1] -32540518.2.2.9 \(s=\left\{Nayibe,Jhon\right\}\)
\[ \Delta_{{Nayibe},{Nayibe}}=\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}} \]
delta33 <- pikl["Nayibe","Nayibe"]-pikl["Nayibe","Nayibe"]*pikl["Nayibe","Nayibe"]
delta33## [1] 0.2496\[ \Delta_{{Nayibe},{Jhon}}=\pi_{{Nayibe},{Jhon}}-\pi_{{Nayibe}}\pi_{{Jhon}} \]
delta35 <- pikl["Nayibe","Jhon"]-pikl["Nayibe","Nayibe"]*pikl["Jhon","Jhon"]
delta35## [1] -0.0596\[ \Delta_{{Jhon},{Nayibe}}=\pi_{{Jhon},{Nayibe}}-\pi_{{Jhon}}\pi_{{Nayibe}} \]
delta53 <- pikl["Jhon","Nayibe"]-pikl["Jhon","Jhon"]*pikl["Nayibe","Nayibe"]
delta53## [1] -0.0596\[ \Delta_{{Jhon},{Jhon}}=\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}} \]
delta55 <- pikl["Jhon","Jhon"]-pikl["Jhon","Jhon"]*pikl["Jhon","Jhon"]
delta55## [1] 0.1971\[ \frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}=\frac{\pi_{{Nayibe},{Nayibe}}-\pi_{{Nayibe}}\pi_{{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}} \]
delta33/pikl["Nayibe","Nayibe"]## [1] 0.52\[ \frac{\Delta_{{Nayibe},{Jhon}}}{\pi_{{Nayibe},{Jhon}}}=\frac{\pi_{{Nayibe},{Jhon}}-\pi_{{Nayibe}}\pi_{{Jhon}}}{\pi_{{Nayibe},{Jhon}}} \]
delta35/pikl["Nayibe","Jhon"]## [1] -0.8514286\[ \frac{\Delta_{{Jhon},{Nayibe}}}{\pi_{{Jhon},{Nayibe}}}=\frac{\pi_{{Jhon},{Nayibe}}-\pi_{{Jhon}}\pi_{{Nayibe}}}{\pi_{{Jhon},{Nayibe}}} \]
delta53/pikl["Jhon","Nayibe"]## [1] -0.8514286\[ \frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}=\frac{\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}}}{\pi_{{Jhon},{Jhon}}} \]
delta55/pikl["Jhon","Jhon"]## [1] 0.73\[y_{Nayibe}\]
y3 <- y[3]
y3## [1] 5454\[y_{Jhon}\]
y5 <- y[5]
y5## [1] 656\[\breve{y}_{Nayibe}=\frac{y_{Nayibe}}{\pi_{Nayibe}}\]
y3/pik[3]## [1] 11362.5\[\breve{y}_{Jhon}=\frac{y_{Jhon}}{\pi_{Jhon}}\]
y5/pik[5]## [1] 2429.63\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}}+2\frac{\Delta_{{Nayibe},{Jhon}}}{\pi_{{Nayibe},{Jhon}}}\frac{y_{Nayibe}}{\pi_{Nayibe}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}} \end{align} \]
Vtypi <- delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2+2*delta35/pikl["Nayibe","Jhon"]*(y3/pik[3])*(y5/pik[5])+delta55/pikl["Jhon","Jhon"]*(y5/pik[5])**2
Vtypi## [1] 244343858.2.2.10 \(s=\left\{Raul,Jhon\right\}\)
\[ \Delta_{{Raul},{Raul}}=\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}} \]
delta44 <- pikl["Raul","Raul"]-pikl["Raul","Raul"]*pikl["Raul","Raul"]
delta44## [1] 0.2211\[ \Delta_{{Raul},{Jhon}}=\pi_{{Raul},{Jhon}}-\pi_{{Raul}}\pi_{{Jhon}} \]
delta45 <- pikl["Raul","Jhon"]-pikl["Raul","Raul"]*pikl["Jhon","Jhon"]
delta45## [1] -0.0091\[ \Delta_{{Jhon},{Raul}}=\pi_{{Jhon},{Raul}}-\pi_{{Jhon}}\pi_{{Raul}} \]
delta54 <- pikl["Jhon","Raul"]-pikl["Jhon","Jhon"]*pikl["Raul","Raul"]
delta54## [1] -0.0091\[ \Delta_{{Jhon},{Jhon}}=\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}} \]
delta55 <- pikl["Jhon","Jhon"]-pikl["Jhon","Jhon"]*pikl["Jhon","Jhon"]
delta55## [1] 0.1971\[ \frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}=\frac{\pi_{{Raul},{Raul}}-\pi_{{Raul}}\pi_{{Raul}}}{\pi_{{Raul},{Raul}}} \]
delta44/pikl["Raul","Raul"]## [1] 0.67\[ \frac{\Delta_{{Raul},{Jhon}}}{\pi_{{Raul},{Jhon}}}=\frac{\pi_{{Raul},{Jhon}}-\pi_{{Raul}}\pi_{{Jhon}}}{\pi_{{Raul},{Jhon}}} \]
delta45/pikl["Raul","Jhon"]## [1] -0.11375\[ \frac{\Delta_{{Jhon},{Raul}}}{\pi_{{Jhon},{Raul}}}=\frac{\pi_{{Jhon},{Raul}}-\pi_{{Jhon}}\pi_{{Raul}}}{\pi_{{Jhon},{Raul}}} \]
delta54/pikl["Jhon","Raul"]## [1] -0.11375\[ \frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}=\frac{\pi_{{Jhon},{Jhon}}-\pi_{{Jhon}}\pi_{{Jhon}}}{\pi_{{Jhon},{Jhon}}} \]
delta55/pikl["Jhon","Jhon"]## [1] 0.73\[y_{Raul}\]
y4 <- y[4]
y4## [1] 646\[y_{Jhon}\]
y5 <- y[5]
y5## [1] 656\[\breve{y}_{Raul}=\frac{y_{Raul}}{\pi_{Raul}}\]
y4/pik[4]## [1] 1957.576\[\breve{y}_{Jhon}=\frac{y_{Jhon}}{\pi_{Jhon}}\]
y5/pik[5]## [1] 2429.63\[ \begin{align} \widehat{V}[\widehat{t}_{y,\pi}]&=\frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}\frac{y_{Raul}^{2}}{\pi_{Raul}^{2}}+2\frac{\Delta_{{Raul},{Jhon}}}{\pi_{{Raul},{Jhon}}}\frac{y_{Raul}}{\pi_{Raul}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}} \end{align} \]
Vtypi <- delta44/pikl["Raul","Raul"]*(y4/pik[4])**2+2*delta45/pikl["Raul","Jhon"]*(y4/pik[4])*(y5/pik[5])+delta55/pikl["Jhon","Jhon"]*(y5/pik[5])**2
Vtypi## [1] 57947408.2.3 \(cve\left(\widehat{t}_{y,\pi}\right)=\frac{\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}}{\widehat{t}_{y,\pi}}\)
8.2.3.1 \(s=\left\{Santiago,Nestor\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Nestor}}}{\pi_{{Santiago},{Nestor}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}}}{\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Nestor}}{\pi_{Nestor}}} \end{align} \]
cvetypi <- sqrt(delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta12/pikl["Santiago","Nestor"]*(y1/pik[1])*(y2/pik[2])+delta55/pikl["Nestor","Nestor"]*(y2/pik[2])**2)/(y[1]/pik[1]+y[2]/pik[2])
cvetypi## [1] 0.70269738.2.3.2 \(s=\left\{Santiago,Nayibe\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Nayibe}}}{\pi_{{Santiago},{Nayibe}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}}}}{\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Nayibe}}{\pi_{Nayibe}}} \end{align} \]
cvetypi <- sqrt(delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta13/pikl["Santiago","Nayibe"]*(y1/pik[1])*(y3/pik[3])+delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2)/(y[1]/pik[1]+y[3]/pik[3])
cvetypi## [1] 0.71500098.2.3.3 \(s=\left\{Santiago,Raul\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Nestor}}}{\pi_{{Santiago},{Nestor}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}}}{\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Nestor}}{\pi_{Nestor}}} \end{align} \]
cvetypi <- sqrt(delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta14/pikl["Santiago","Raul"]*(y1/pik[1])*(y4/pik[4])+delta44/pikl["Raul","Raul"]*(y4/pik[4])**2)/(y[1]/pik[1]+y[4]/pik[4])
cvetypi## [1] 0.78700148.2.3.4 \(s=\left\{Santiago,Jhon\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Jhon}}}{\pi_{{Santiago},{Jhon}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}}}}{\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Jhon}}{\pi_{Jhon}}} \end{align} \]
cvetypi <- sqrt(delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta15/pikl["Santiago","Jhon"]*(y1/pik[1])*(y5/pik[5])+delta55/pikl["Raul","Raul"]*(y5/pik[5])**2)/(y[1]/pik[1]+y[5]/pik[5])
cvetypi## [1] 0.7393748.2.3.5 \(s=\left\{Nestor,Nayibe\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}+2\frac{\Delta_{{Nestor},{Nayibe}}}{\pi_{{Nestor},{Nayibe}}}\frac{y_{Nestor}}{\pi_{Nestor}}\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}}}}{\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{y_{Nayibe}}{\pi_{Nayibe}}} \end{align} \]
cvetypi <- sqrt(delta22/pikl["Nestor","Nestor"]*(y2/pik[2])**2+2*delta23/pikl["Nestor","Nayibe"]*(y2/pik[2])*(y3/pik[3])+delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2)/(y[2]/pik[2]+y[3]/pik[3])
cvetypi## [1] 0.63988398.2.3.6 \(s=\left\{Nestor,Raul\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}+2\frac{\Delta_{{Nestor},{Raul}}}{\pi_{{Nestor},{Raul}}}\frac{y_{Nestor}}{\pi_{Nestor}}\frac{y_{Raul}}{\pi_{Raul}}+\frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}\frac{y_{Raul}^{2}}{\pi_{Raul}^{2}}}}{\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{y_{Raul}}{\pi_{Raul}}} \end{align} \]
cvetypi <- sqrt(delta22/pikl["Nestor","Nestor"]*(y2/pik[2])**2+2*delta24/pikl["Nestor","Raul"]*(y2/pik[2])*(y4/pik[4])+delta44/pikl["Raul","Raul"]*(y4/pik[4])**2)/(y[2]/pik[2]+y[4]/pik[4])## Warning in sqrt(delta22/pikl["Nestor", "Nestor"] * (y2/pik[2])^2 + 2 * delta24/
## pikl["Nestor", : Se han producido NaNscvetypi## [1] NaN8.2.3.7 \(s=\left\{Nestor,Jhon\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}+2\frac{\Delta_{{Nestor},{Jhon}}}{\pi_{{Nestor},{Jhon}}}\frac{y_{Nestor}}{\pi_{Nestor}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}}}}{\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{y_{Jhon}}{\pi_{Jhon}}} \end{align} \]
cvetypi <- sqrt(delta22/pikl["Nestor","Nestor"]*(y2/pik[2])**2+2*delta25/pikl["Nestor","Jhon"]*(y2/pik[2])*(y5/pik[5])+delta44/pikl["Jhon","Jhon"]*(y5/pik[5])**2)/(y[2]/pik[2]+y[5]/pik[5])## Warning in sqrt(delta22/pikl["Nestor", "Nestor"] * (y2/pik[2])^2 + 2 * delta25/
## pikl["Nestor", : Se han producido NaNscvetypi## [1] NaN8.2.3.8 \(s=\left\{Nayibe,Raul\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}}+2\frac{\Delta_{{Nayibe},{Raul}}}{\pi_{{Nayibe},{Raul}}}\frac{y_{Nayibe}}{\pi_{Nayibe}}\frac{y_{Raul}}{\pi_{Raul}}+\frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}\frac{y_{Raul}^{2}}{\pi_{Raul}^{2}}}}{\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{y_{Raul}}{\pi_{Raul}}} \end{align} \]
cvetypi <- sqrt(delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2+2*delta34/pikl["Nayibe","Raul"]*(y3/pik[3])*(y4/pik[4])+delta44/pikl["Raul","Raul"]*(y4/pik[4])**2)/(y[3]/pik[3]+y[4]/pik[4])## Warning in sqrt(delta33/pikl["Nayibe", "Nayibe"] * (y3/pik[3])^2 + 2 * delta34/
## pikl["Nayibe", : Se han producido NaNscvetypi## [1] NaN8.2.3.9 \(s=\left\{Nayibe,Jhon\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Nayibe},{Nayibe}}}{\pi_{{Nayibe},{Nayibe}}}\frac{y_{Nayibe}^{2}}{\pi_{Nayibe}^{2}}+2\frac{\Delta_{{Nayibe},{Jhon}}}{\pi_{{Nayibe},{Jhon}}}\frac{y_{Nayibe}}{\pi_{Nayibe}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}}}}{\frac{y_{Nayibe}}{\pi_{Nayibe}}+\frac{y_{Jhon}}{\pi_{Jhon}}} \end{align} \]
cvetypi <- sqrt(delta33/pikl["Nayibe","Nayibe"]*(y3/pik[3])**2+2*delta35/pikl["Nayibe","Jhon"]*(y3/pik[3])*(y5/pik[5])+delta55/pikl["Jhon","Jhon"]*(y5/pik[5])**2)/(y[3]/pik[3]+y[5]/pik[5])
cvetypi## [1] 0.35840118.2.3.10 \(s=\left\{Raul,Jhon\right\}\)
\[ \begin{align} cve\left(\widehat{t}_{y,\pi}\right)&=\frac{\sqrt{\frac{\Delta_{{Raul},{Raul}}}{\pi_{{Raul},{Raul}}}\frac{y_{Raul}^{2}}{\pi_{Raul}^{2}}+2\frac{\Delta_{{Raul},{Jhon}}}{\pi_{{Raul},{Jhon}}}\frac{y_{Raul}}{\pi_{Raul}}\frac{y_{Jhon}}{\pi_{Jhon}}+\frac{\Delta_{{Jhon},{Jhon}}}{\pi_{{Jhon},{Jhon}}}\frac{y_{Jhon}^{2}}{\pi_{Jhon}^{2}}}}{\frac{y_{Raul}}{\pi_{Raul}}+\frac{y_{Jhon}}{\pi_{Jhon}}} \end{align} \]
cvetypi <- sqrt(delta44/pikl["Raul","Raul"]*(y3/pik[3])**2+2*delta45/pikl["Raul","Jhon"]*(y4/pik[4])*(y5/pik[5])+delta55/pikl["Jhon","Jhon"]*(y5/pik[5])**2)/(y[4]/pik[4]+y[5]/pik[5])
cvetypi## [1] 2.1591238.2.4 \(ci_{1-\alpha}\left({\widehat{t}_{y,\pi}}\right)=\widehat{t}_{y,\pi}{\pm}{Z}_{1-\frac{\alpha}{2}}\sqrt{\widehat{V}[\widehat{t}_{y,\pi}]}\)
8.2.4.1 \(s=\left\{Santiago,Nestor\right\}\)
\[ \begin{align} ci_{1-\alpha}\left({\widehat{t}_{y,\pi}}\right)&=\frac{y_{Santiago}}{\pi_{Santiago}}+\frac{y_{Nestor}}{\pi_{Nestor}}{\pm}{Z}_{1-\frac{\alpha}{2}}\sqrt{\frac{\Delta_{{Santiago},{Santiago}}}{\pi_{{Santiago},{Santiago}}}\frac{y_{Santiago}^{2}}{\pi_{Santiago}^{2}}+2\frac{\Delta_{{Santiago},{Nestor}}}{\pi_{{Santiago},{Nestor}}}\frac{y_{Santiago}}{\pi_{Santiago}}\frac{y_{Nestor}}{\pi_{Nestor}}+\frac{\Delta_{{Nestor},{Nestor}}}{\pi_{{Nestor},{Nestor}}}\frac{y_{Nestor}^{2}}{\pi_{Nestor}^{2}}} \end{align} \]
cvetypi <- (y[1]/pik[1]+y[2]/pik[2])+c(qnorm(0.025)*sqrt(delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta12/pikl["Santiago","Nestor"]*(y1/pik[1])*(y2/pik[2])+delta55/pikl["Nestor","Nestor"]*(y2/pik[2])**2),qnorm(0.975)*sqrt(delta11/pikl["Santiago","Santiago"]*(y1/pik[1])**2+2*delta12/pikl["Santiago","Nestor"]*(y1/pik[1])*(y2/pik[2])+delta55/pikl["Nestor","Nestor"]*(y2/pik[2])**2))
cvetypi## [1] -511.2544 3221.59928.3 Muestras representativas
8.4 Estrategias representativas
9 Muestras con probabilidades simples
9.1 Diseño de muestreo Bernoulli
\[ \mathcal{p}(s)= \begin{cases} {\pi}^{n{(s)}}\left({1}-{\pi}\right)^{N-n{(s)}}&\text{ si }s\text{ tiene tamaño igual a }{n{(s)}}\\ {0}&\text{ en cualquier otro caso} \end{cases} \]
9.1.1 Algoritmo de selección
\({0}<{\pi}<{1}\)
\(\forall_{k{\in}U}{\varepsilon}_{k}{\sim}U{\left[0,1\right]}\)
\({\varepsilon}_{k}<{\pi}{\implies}k{\in}s\)
\[\forall_{k{\in}U}\mathcal{P}\left({\varepsilon}_{k}<{\pi}\right)={\pi}{\implies}I_k(S){\sim}Bernoulli(\pi)\]
\[Q_r=\{s{\in}Q\mid\#(s)=r\}\]
\[ \begin{align} \#\left(Q_r\right)&=\binom{N}{r}\\ &=\frac{N!}{(N-r)!r!} \end{align} \]
\[Q=\{s{\in}Q\mid\#(s)=0,\ldots,r\}\]
\[ \begin{align} \#\left(Q\right)&={\sum}_{r=0}^{N}\binom{N}{r}\\ &={\sum}_{r=0}^{N}\binom{N}{r}{1}^{N}\\ &={\sum}_{r=0}^{N}\binom{N}{r}{1}^{N-r+r}\\ &={\sum}_{r=0}^{N}\binom{N}{r}{1}^{r}{1}^{N-r}\\ &={(1+1)}^{N}\\ &={2}^{N} \end{align} \]
\[ \begin{align} \mathcal{P}\left[n(s)=r\right]&={\sum}_{s{\in}Qr}p(s)\\ &=\binom{N}{r}{\pi}^{r}\left({1}-{\pi}\right)^{N-r} \end{align} \]
\[{\forall}_{r=1,\ldots,N}{Qr}{\subset}{Q}\]
\[E\left[n{\left(S\right)}\right]=N{\pi}\]
\[V\left[n{\left(S\right)}\right]=N{\pi}{\left(1-{\pi}\right)}\]
\[ \begin{align} {\sum}_{s{\in}Q}\mathcal{p}\left(s\right)&={\sum}_{s{\in}Q\&\#{(s)}={0}}\mathcal{p}\left(s\right)+{\sum}_{s{\in}Q\&\#{(s)}={1}}\mathcal{p}\left(s\right)+\cdots+{\sum}_{s{\in}Q\&\#{(s)}={N}}\mathcal{p}\left(s\right)\\ &=\binom{N}{0}{\pi}^{0}\left({1}-{\pi}\right)^{N-0}+\binom{N}{1}{\pi}^{1}\left({1}-{\pi}\right)^{N-1}+\cdots+\binom{N}{N}{\pi}^{N}\left({1}-{\pi}\right)^{N-N}\\ &={\sum}_{r=0}^{N}\binom{N}{r}{\pi}^{r}\left({1}-{\pi}\right)^{N-r}\\ &=\left[{\pi}+\left({1}-{\pi}\right)\right]^{N}\\ &={1}^{N}\\ &={1} \end{align} \] \[ \begin{align} \pi_k&={\sum}_{s{\in}Q}I_k\left({s}\right)\mathcal{p}\left({s}\right)\\ &=0\mathcal{p}{(s_0)}+\binom{1}{1}\binom{N-1}{0}\mathcal{p}{(s_1)}+\cdots+\binom{1}{1}\binom{N-1}{N-1}\mathcal{p}{(s_N)}\\ &=\binom{N-1}{0}\mathcal{p}{(s_1)}+\cdots+\binom{N-1}{N-1}\mathcal{p}{(s_N)}\\ &={\sum}_{r=0}^{N-1}\binom{N-1}{r}\mathcal{p}{(s_r)}\\ &={\sum}_{r=0}^{N-1}\binom{N-1}{r}{\pi}^{r+1}({1-{\pi}})^{N-(r+1)}\\ &={\sum}_{r=0}^{N-1}\binom{N-1}{r}{\pi}^{r+1}({1-{\pi}})^{N-r-1}\\ &=\pi{\sum}_{r=0}^{N-1}\binom{N-1}{r}{\pi}^{r}({1-{\pi}})^{N-r}\\ &=\pi\left[{\pi+(1-\pi)}\right]^{N-1}\\ &=\pi\left[{\pi+1-\pi}\right]^{N-1}\\ &=\pi\left[{1}\right]^{N-1}\\ &=\pi \end{align} \]
\[ \begin{align} \mathcal{P}(k{\in}S\text{ & }l{\in}S)&=\mathcal{P}(I_k=1)\mathcal{P}(I_l=1)\\ &={\pi}{\cdot}{\pi}\\ &={\pi}^{2} \end{align} \]
9.1.2 Estimador de Horvitz - Thompson
\[ \begin{align} \widehat{t}_{y,\pi}&={\sum}_{s}\frac{y_k}{\pi}\\ &=\frac{1}{\pi}{\sum}_{s}y_k \end{align} \]
\[ \begin{align} V[\widehat{t}_{y,\pi}]&={{\sum}{\sum}}_{U}{\Delta}_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l} \end{align} \]
\[ \begin{align} {\Delta}_{kl}&= \begin{cases} {\pi}_{kl}-{\pi}_{k}{\pi}_{l}&\text{ para }k{\neq}l\\ {\pi}_{k}-{\pi}_{k}^{2}&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} {\pi}{\pi}-{\pi}^{2}&\text{ para }k{\neq}l\\ {\pi}-{\pi}^{2}&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} {\pi}^{2}-{\pi}^{2}&\text{ para }k{\neq}l\\ {\pi}\left(1-{\pi}\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} {0}&\text{ para }k{\neq}l\\ {\pi}\left(1-{\pi}\right)&\text{ para }k{=}l \end{cases} \end{align} \]
\[ \begin{align} V_{BER}[\widehat{t}_{y,\pi}]&={\sum}_{U}{{\pi}\left(1-{\pi}\right)}\frac{y_k}{\pi}\frac{y_k}{\pi}\\ &={\sum}_{U}{{\pi}\left(1-{\pi}\right)}\frac{y_k^2}{{\pi}^{2}}\\ &={\sum}_{U}\frac{{\pi}\left(1-{\pi}\right)}{{\pi}^{2}}{y}_{k}^{2}\\ &={\sum}_{U}\frac{1-{\pi}}{\pi}{y}_{k}^{2}\\ &={\sum}_{U}\left(\frac{1}{\pi}-{1}\right){y}_{k}^{2}\\ &=\left(\frac{1}{\pi}-{1}\right){\sum}_{U}{y}_{k}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{BER}[\widehat{t}_{y,\pi}]&={\sum}_{s}\frac{{\pi}\left(1-{\pi}\right)}{\pi}\frac{y_k}{\pi}\frac{y_k}{\pi}\\ &={\sum}_{s}\frac{{\pi}\left(1-{\pi}\right)}{\pi}\frac{y_k^2}{{\pi}^{2}}\\ &={\sum}_{s}\frac{{\pi}\left(1-{\pi}\right)}{{\pi}^{3}}{y}_{k}^{2}\\ &={\sum}_{s}\frac{1-{\pi}}{{\pi}^{2}}{y}_{k}^{2}\\ &={\sum}_{s}\frac{1}{\pi}\left(\frac{1}{\pi}-{1}\right){y}_{k}^{2}\\ &=\frac{1}{\pi}\left(\frac{1}{\pi}-{1}\right){\sum}_{s}{y}_{k}^{2} \end{align} \]
\[ \begin{align} \widehat{t}_{y,\pi}(s=U)&={\sum}_{s=U}\frac{y_k}{\pi}\\ &=\frac{1}{\pi}{\sum}_{s=U}y_k\\ &=\frac{{t}_{y}}{\pi}\\ &\neq{t}_{y} \end{align} \]
\[ \begin{align} \widehat{t}_{y,alt}&=N\tilde{y}_{s}\\ &=N\frac{{\sum}_{s}\frac{y_k}{\pi}}{{\sum}_{s}\frac{1}{\pi}}\\ &=N\frac{\widehat{t}_{y\pi}}{\widehat{N}}\\ &=N\frac{\widehat{t}_{y\pi}}{n{(s)}}\\ &=N\tilde{y}_{s}\\ \end{align} \]
attach(Lucy)
N <- dim(Lucy)[1];N## [1] 2396pik <- 400/N;pik## [1] 0.1669449seleccion <- S.BE(N,pik)
muestra <- Lucy[seleccion,]
attach(muestra);muestra## ID Ubication Level Zone Income Employees Taxes SPAM
## 5 AB005 c1k5 Small A 391 91 7.0 yes
## 8 AB008 c1k8 Small A 473 57 10.0 yes
## 13 AB013 c1k13 Small A 402 19 7.0 yes
## 14 AB014 c1k14 Small A 330 23 4.0 yes
## 19 AB019 c1k19 Small A 342 60 5.0 yes
## 29 AB029 c1k29 Small A 310 94 4.0 yes
## 31 AB031 c1k31 Small A 378 94 6.0 yes
## 32 AB032 c1k32 Small A 380 18 6.0 yes
## 48 AB048 c1k48 Small A 422 101 8.0 yes
## 56 AB059 c1k56 Small A 350 68 5.0 yes
## 58 AB061 c1k58 Small A 363 73 6.0 no
## 63 AB066 c1k63 Small A 340 28 5.0 yes
## 65 AB068 c1k65 Small A 360 61 5.0 no
## 69 AB072 c1k69 Small A 390 95 7.0 yes
## 73 AB078 c1k73 Small A 330 79 4.0 yes
## 74 AB079 c1k74 Small A 401 39 7.0 no
## 90 AB106 c1k90 Small A 450 87 9.0 yes
## 96 AB116 c1k96 Small A 480 53 10.0 no
## 101 AB1219 c2k2 Small B 437 41 8.0 yes
## 123 AB1333 c2k24 Small B 312 82 4.0 no
## 128 AB1338 c2k29 Small B 282 69 3.0 yes
## 136 AB1345 c2k37 Small B 330 47 4.0 yes
## 142 AB1350 c2k43 Small B 280 69 3.0 no
## 145 AB1353 c2k46 Small B 240 83 2.0 yes
## 146 AB1354 c2k47 Small B 310 42 4.0 no
## 148 AB1356 c2k49 Small B 315 75 4.0 yes
## 152 AB136 c2k53 Small B 75 21 0.5 yes
## 159 AB1366 c2k60 Small B 230 10 2.0 yes
## 160 AB1367 c2k61 Small B 200 49 1.0 yes
## 166 AB1372 c2k67 Small B 318 51 4.0 yes
## 192 AB1396 c2k93 Small B 325 43 4.0 yes
## 195 AB1399 c2k96 Small B 230 23 2.0 yes
## 200 AB1403 c3k2 Small B 310 54 4.0 yes
## 205 AB1408 c3k7 Small B 263 56 3.0 yes
## 206 AB1409 c3k8 Small B 261 56 2.0 no
## 209 AB1411 c3k11 Small B 300 70 3.0 no
## 211 AB1413 c3k13 Small B 199 49 1.0 no
## 220 AB1421 c3k22 Small B 370 30 6.0 no
## 221 AB1422 c3k23 Small B 295 57 3.0 no
## 226 AB1427 c3k28 Small B 300 57 3.0 yes
## 233 AB1433 c3k35 Small B 211 26 1.0 yes
## 244 AB1443 c3k46 Small B 310 78 4.0 yes
## 248 AB1447 c3k50 Small B 340 24 5.0 yes
## 252 AB1450 c3k54 Small B 334 80 5.0 no
## 253 AB1451 c3k55 Small B 232 47 2.0 no
## 255 AB1453 c3k57 Small B 290 57 3.0 yes
## 262 AB146 c3k64 Small B 82 13 0.5 no
## 263 AB1460 c3k65 Small B 300 78 3.0 yes
## 282 AB1478 c3k84 Small B 267 88 3.0 no
## 286 AB1481 c3k88 Small B 220 86 2.0 no
## 294 AB1489 c3k96 Small B 333 80 5.0 no
## 325 AB1517 c4k28 Small B 208 22 1.0 no
## 328 AB152 c4k31 Small B 193 81 1.0 no
## 341 AB1531 c4k44 Small B 295 49 3.0 yes
## 343 AB1533 c4k46 Small B 344 24 5.0 yes
## 348 AB1538 c4k51 Small B 280 13 3.0 yes
## 354 AB1543 c4k57 Small B 318 71 4.0 yes
## 355 AB1544 c4k58 Small B 245 63 2.0 yes
## 360 AB1549 c4k63 Small B 218 54 2.0 no
## 364 AB1552 c4k67 Small B 227 6 2.0 yes
## 370 AB1558 c4k73 Small B 206 10 1.0 yes
## 378 AB1565 c4k81 Small B 206 46 1.0 no
## 383 AB157 c4k86 Small B 185 52 1.0 yes
## 385 AB1571 c4k88 Small B 290 45 3.0 yes
## 392 AB1578 c4k95 Small B 315 31 4.0 no
## 393 AB1579 c4k96 Small B 238 71 2.0 yes
## 394 AB158 c4k97 Small B 76 85 0.5 yes
## 395 AB1580 c4k98 Small B 299 85 3.0 no
## 401 AB1586 c5k5 Small B 261 40 2.0 no
## 410 AB1594 c5k14 Small B 264 44 3.0 no
## 412 AB1596 c5k16 Small B 243 35 2.0 no
## 415 AB1599 c5k19 Small B 268 44 3.0 no
## 425 AB1608 c5k29 Small B 303 94 4.0 yes
## 426 AB1609 c5k30 Small B 260 32 2.0 yes
## 429 AB1611 c5k33 Small B 213 38 1.0 no
## 440 AB1621 c5k44 Small B 276 57 3.0 no
## 452 AB1632 c5k56 Small B 281 85 3.0 no
## 456 AB1636 c5k60 Small B 251 23 2.0 no
## 458 AB1638 c5k62 Small B 264 40 3.0 no
## 462 AB1641 c5k66 Small B 217 58 2.0 yes
## 481 AB1659 c5k85 Small B 278 53 3.0 yes
## 484 AB1661 c5k88 Small B 217 6 2.0 yes
## 486 AB1663 c5k90 Small B 292 57 3.0 no
## 487 AB1664 c5k91 Small B 273 56 3.0 yes
## 495 AB1671 c5k99 Small B 231 23 2.0 yes
## 508 AB1683 c6k13 Small B 332 92 4.0 no
## 511 AB1686 c6k16 Small B 269 36 3.0 yes
## 512 AB1687 c6k17 Small B 315 63 4.0 yes
## 516 AB1690 c6k21 Small B 180 64 1.0 no
## 518 AB1692 c6k23 Small B 130 14 0.5 yes
## 519 AB1693 c6k24 Small B 169 84 1.0 yes
## 530 AB1703 c6k35 Small B 230 75 2.0 no
## 535 AB1708 c6k40 Small B 154 7 1.0 yes
## 539 AB1711 c6k44 Small B 180 12 1.0 yes
## 542 AB1714 c6k47 Small B 130 50 0.5 no
## 550 AB1721 c6k55 Small B 142 67 0.5 no
## 553 AB1724 c6k58 Small B 195 45 1.0 no
## 557 AB1728 c6k62 Small B 145 23 0.5 yes
## 560 AB1730 c6k65 Small B 200 65 1.0 no
## 568 AB1738 c6k73 Small B 1 13 0.5 no
## 578 AB1747 c6k83 Small B 150 35 1.0 yes
## 588 AB1756 c6k93 Small B 141 27 0.5 yes
## 598 AB1765 c7k4 Small B 226 78 2.0 no
## 600 AB1767 c7k6 Small B 175 44 1.0 yes
## 608 AB1774 c7k14 Small B 188 81 1.0 yes
## 619 AB1784 c7k25 Small B 182 76 1.0 yes
## 627 AB1791 c7k33 Small B 97 21 0.5 yes
## 628 AB1792 c7k34 Small B 96 17 0.5 yes
## 635 AB1799 c7k41 Small B 130 38 0.5 yes
## 638 AB1801 c7k44 Small B 130 30 0.5 yes
## 656 AB1818 c7k62 Small B 151 7 1.0 yes
## 658 AB182 c7k64 Small B 87 5 0.5 yes
## 666 AB1827 c7k72 Small B 160 56 1.0 yes
## 681 AB1840 c7k87 Small B 185 64 1.0 yes
## 691 AB185 c7k97 Small C 68 73 0.5 yes
## 694 AB1852 c8k1 Small C 166 40 1.0 yes
## 695 AB1853 c8k2 Small C 190 21 1.0 yes
## 697 AB1855 c8k4 Small C 131 74 0.5 yes
## 716 AB1872 c8k23 Small C 136 23 0.5 no
## 717 AB1873 c8k24 Small C 171 60 1.0 yes
## 726 AB1881 c8k33 Small C 450 54 9.0 no
## 736 AB1890 c8k43 Small C 450 55 9.0 no
## 749 AB1902 c8k56 Small C 480 93 10.5 no
## 752 AB1906 c8k59 Small C 460 43 9.0 no
## 775 AB1929 c8k82 Small C 410 16 7.0 yes
## 782 AB1936 c8k89 Small C 310 18 4.0 no
## 797 AB1951 c9k5 Small C 480 33 10.0 yes
## 804 AB1958 c9k12 Small C 490 94 10.5 no
## 806 AB1960 c9k14 Small C 312 74 4.0 yes
## 808 AB1962 c9k16 Small C 379 86 6.0 yes
## 813 AB1967 c9k21 Small C 487 25 10.5 no
## 819 AB1972 c9k27 Small C 400 31 7.0 yes
## 821 AB1974 c9k29 Small C 410 28 7.0 yes
## 822 AB1975 c9k30 Small C 420 36 8.0 yes
## 828 AB1980 c9k36 Small C 363 45 5.0 yes
## 831 AB1983 c9k39 Small C 440 94 8.0 yes
## 839 AB1991 c9k47 Small C 390 31 7.0 no
## 843 AB1995 c9k51 Small C 407 64 7.0 no
## 864 AB202 c9k72 Small C 129 46 0.5 yes
## 867 AB2023 c9k75 Small C 469 20 10.0 no
## 868 AB2024 c9k76 Small C 390 23 6.0 yes
## 870 AB2026 c9k78 Small C 460 39 9.0 yes
## 888 AB2045 c9k96 Small C 440 34 8.0 yes
## 892 AB2050 c10k1 Small C 425 37 8.0 no
## 897 AB2058 c10k6 Small C 480 93 10.0 no
## 899 AB206 c10k8 Small C 110 62 0.5 no
## 901 AB2062 c10k10 Small C 333 76 5.0 no
## 903 AB2064 c10k12 Small C 420 76 8.0 yes
## 906 AB2068 c10k15 Small C 360 49 5.0 yes
## 916 AB2077 c10k25 Small C 385 38 6.0 yes
## 921 AB2082 c10k30 Small C 480 41 10.0 yes
## 926 AB209 c10k35 Small C 96 49 0.5 no
## 932 AB2097 c10k41 Small C 470 36 10.0 no
## 933 AB2098 c10k42 Small C 494 102 10.5 no
## 935 AB210 c10k44 Small C 117 46 0.5 no
## 942 AB2107 c10k51 Small C 469 40 10.0 no
## 949 AB2116 c10k58 Small C 457 95 9.0 no
## 951 AB2118 c10k60 Small C 480 85 10.0 no
## 956 AB2122 c10k65 Small C 420 28 7.0 no
## 961 AB2129 c10k70 Small C 390 43 6.0 no
## 962 AB213 c10k71 Small C 186 80 1.0 no
## 964 AB2131 c10k73 Small C 334 76 5.0 yes
## 965 AB2132 c10k74 Small C 380 22 6.0 no
## 979 AB2147 c10k88 Small C 487 105 10.5 yes
## 987 AB2155 c10k96 Small C 480 41 10.0 no
## 998 AB2165 c11k8 Small C 340 68 5.0 yes
## 1002 AB2169 c11k12 Small C 260 68 2.0 no
## 1009 AB2175 c11k19 Small C 420 40 7.0 no
## 1010 AB2178 c11k20 Small C 494 38 10.5 no
## 1013 AB2182 c11k23 Small C 370 46 6.0 yes
## 1015 AB2184 c11k25 Small C 480 53 10.0 no
## 1016 AB2186 c11k26 Small C 345 40 5.0 yes
## 1026 AB2198 c11k36 Small C 366 69 6.0 no
## 1028 AB220 c11k38 Small C 182 24 1.0 yes
## 1031 AB2202 c11k41 Small C 378 70 6.0 yes
## 1049 AB2220 c11k59 Small C 402 39 7.0 yes
## 1065 AB2239 c11k75 Small C 368 17 6.0 no
## 1070 AB2243 c11k80 Small C 491 42 10.5 yes
## 1071 AB2244 c11k81 Small C 491 94 10.5 no
## 1073 AB2246 c11k83 Small C 460 76 9.0 yes
## 1075 AB2248 c11k85 Small C 411 52 7.0 yes
## 1076 AB2249 c11k86 Small C 373 50 6.0 no
## 1079 AB2252 c11k89 Small C 341 24 5.0 yes
## 1080 AB2254 c11k90 Small C 470 40 10.0 no
## 1088 AB2262 c11k98 Small C 460 55 9.0 yes
## 1090 AB2265 c12k1 Small C 477 61 10.0 yes
## 1094 AB227 c12k5 Small C 122 54 0.5 yes
## 1097 AB2273 c12k8 Small C 357 21 5.0 no
## 1101 AB2277 c12k12 Small C 370 33 6.0 yes
## 1109 AB2285 c12k20 Small C 428 57 8.0 no
## 1111 AB2287 c12k22 Small C 499 90 10.5 yes
## 1119 AB2294 c12k30 Small C 460 27 9.0 no
## 1124 AB2299 c12k35 Small C 324 31 4.0 no
## 1125 AB230 c12k36 Small C 158 15 1.0 yes
## 1133 AB231 c12k44 Small C 42 57 0.5 no
## 1136 AB2312 c12k47 Small C 424 61 8.0 yes
## 1138 AB2314 c12k49 Small C 451 19 9.0 no
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## 1750 AB1181 c18k67 Medium A 810 120 32.0 no
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## 1761 AB1192 c18k78 Medium B 552 95 14.0 yes
## 1772 AB1205 c18k89 Medium B 675 100 23.0 no
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## 1789 AB1234 c19k7 Medium B 704 47 25.0 yes
## 1790 AB1235 c19k8 Medium B 530 97 13.0 no
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## 1814 AB1263 c19k32 Medium B 610 46 18.0 no
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## 1819 AB1268 c19k37 Medium B 580 58 16.0 yes
## 1828 AB1277 c19k46 Medium B 821 71 34.0 yes
## 1829 AB1278 c19k47 Medium B 804 103 32.0 no
## 1835 AB1285 c19k53 Medium B 669 79 22.0 yes
## 1836 AB1286 c19k54 Medium B 890 60 39.0 yes
## 1842 AB1292 c19k60 Medium B 675 88 23.0 no
## 1843 AB1294 c19k61 Medium B 605 81 18.0 yes
## 1845 AB1296 c19k63 Medium B 674 47 23.0 yes
## 1852 AB1303 c19k70 Medium B 503 91 12.0 yes
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## 1869 AB1323 c19k87 Medium B 600 96 18.0 no
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## 1873 AB1327 c19k91 Medium B 500 42 11.0 yes
## 1885 AB1987 c20k4 Medium C 525 65 13.0 yes
## 1895 AB2042 c20k14 Medium C 510 80 12.0 yes
## 1901 AB2061 c20k20 Medium C 510 84 12.0 no
## 1909 AB2106 c20k28 Medium C 500 55 11.0 yes
## 1917 AB2154 c20k36 Medium C 504 87 12.0 yes
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## 1961 AB579 c20k80 Medium C 700 97 24.0 yes
## 1970 AB588 c20k89 Medium C 650 41 21.0 yes
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## 1976 AB596 c20k95 Medium C 570 77 15.0 no
## 1977 AB597 c20k96 Medium C 570 73 15.0 yes
## 1978 AB598 c20k97 Medium C 630 111 20.0 yes
## 1980 AB601 c20k99 Medium C 830 135 34.0 no
## 1981 AB602 c22k1 Medium C 840 89 35.0 yes
## 1983 AB605 c22k3 Medium C 619 62 19.0 yes
## 1984 AB606 c22k4 Medium C 500 39 12.0 yes
## 1992 AB615 c22k12 Medium D 920 142 41.0 yes
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## 2384 AB908 c26k8 Big E 1305 150 70.0 yes
## 2385 AB912 c26k9 Big E 1088 133 55.0 no
## 2389 AB956 c26k13 Big E 1040 78 51.0 yes
## 2393 AB985 c26k17 Big E 1077 91 54.0 no
## 2396 AB988 c26k20 Big E 1860 253 176.0 yesa.estimar <- data.frame(Income, Employees, Taxes);E.BE(a.estimar,pik)## N Income Employees Taxes
## Estimation 2581.69000 1.133613e+06 1.641919e+05 32223.205000
## Standard Error 113.50169 5.920796e+04 8.132540e+03 2458.408173
## CVE 4.39641 5.222941e+00 4.953071e+00 7.629310
## DEFF Inf 3.423078e+00 4.702865e+00 1.4936839.2 Muestreo aleatorio simple sin reemplazo
\[ \mathcal{p}(s)= \begin{cases} \frac{1}{\binom{N}{n}}&\text{ si }s\text{ tiene tamaño igual a }{n{(s)}=n}\\ {0}&\text{ en cualquier otro caso} \end{cases} \]
9.2.1 Algoritmos de selección
9.2.1.1 Coordinado negativo
\(\forall_{k{\in}U}{\xi}_{k}{\sim}U{\left(0,1\right)}\)
\({\xi}_{(k)}{\leq}{n}{\implies}k{\in}s\)
9.2.1.2 Selección y rechazo
\(\forall_{k{\in}U}{\xi}_{k}{\sim}U{\left(0,1\right)}\)
\({c}_{k}=\frac{{n}-{n}_{k}}{N-k+1}\)
\({\xi}_{(k)}{\leq}{{c}_{k}}{\implies}k{\in}s\)
\[ \begin{align} \#\left(Q\right)&=\binom{N}{n} \end{align} \]
\[ \begin{align} {\sum}_{s{\in}Q}p(s)&={\sum}_{s{\in}Q}\frac{1}{\binom{N}{n}}\\ &={\sum}_{i=1}^{\binom{N}{n}}\frac{1}{\binom{N}{n}}\\ &=\binom{N}{n}\frac{1}{\binom{N}{n}}\\ &=1 \end{align} \]
\[E\left[n{\left(S\right)}\right]=N{\pi}\]
\[V\left[n{\left(S\right)}\right]=N{\pi}{\left(1-{\pi}\right)}\]
\[ \begin{align} \pi_k&={\sum}_{s{\in}Q}I_k\left({s}\right)\mathcal{p}\left({s}\right)\\ &=\frac{\binom{1}{1}\binom{N-1}{n-1}}{\binom{N}{n}}\\ &=\frac{n}{N} \end{align} \]
\[ \begin{align} \pi_{kl}&=\mathcal{P}(k{\in}S\text{ & }l{\in}S)\\ &=\mathcal{P}(I_k(S)=1{\mid}I_l(S)=1)\mathcal{P}(I_l(S)=1)\\ &=\frac{n-1}{N-1}\frac{n}{N}\\ &=\frac{(n-1)n}{(N-1)N} \end{align} \]
\[ \begin{align} \mathcal{P}(s{=}S\text{ & }n(S){=}n)&=\frac{\mathcal{P}(s{=}S\text{ & }n(S){=}n)}{\mathcal{P}(n(S){=}n)}\\ &=\frac{{\pi}^{n}\left(1-{\pi}\right)^{N-n}}{\binom{N}{n}{\pi}^{n}\left(1-{\pi}\right)^{N-n}}\\ &=\frac{1}{\binom{N}{n}} \end{align} \]
9.2.2 Estimador de Horvitz - Thompson
\[ \begin{align} \widehat{t}_{y,\pi}&={\sum}_{s}\frac{y_k}{\pi_k}\\ &=\frac{1}{\pi_k}{\sum}_{s}y_k\\ &=\frac{1}{\frac{n}{N}}{\sum}_{s}y_k\\ &=\frac{N}{n}{\sum}_{s}y_k \end{align} \]
\[ \begin{align} V[\widehat{t}_{y,\pi}]&={{\sum}{\sum}}_{U}{\Delta}_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l} \end{align} \]
\[ \begin{align} {\Delta}_{kl}&= \begin{cases} {\pi}_{kl}-{\pi}_{k}{\pi}_{l}&\text{ para }k{\neq}l\\ {\pi}_{k}\left(1-{\pi}_{k}\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} \frac{(n-1)n}{(N-1)N}-\frac{n}{N}\frac{n}{N}&\text{ para }k{\neq}l\\ \frac{n}{N}\left(1-\frac{n}{N}\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} \left[\frac{n-1}{N-1}-\frac{n}{N}\right]\frac{n}{N}&\text{ para }k{\neq}l\\ \frac{n}{N}\left(\frac{N}{N}-\frac{n}{N}\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} \left[\frac{N(n-1)-n(N-1)}{N(N-1)}\right]\frac{n}{N}&\text{ para }k{\neq}l\\ \frac{n}{N}\left(\frac{N-n}{N}\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} \left[\frac{Nn-N-nN+n)}{N(N-1)}\right]\frac{n}{N}&\text{ para }k{\neq}l\\ \frac{n}{N^2}\left(N-n\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} \left[\frac{n-N}{N(N-1)}\right]\frac{n}{N}&\text{ para }k{\neq}l\\ \frac{n}{N^2}\left(N-n\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} -\frac{n}{N^2}\left[\frac{N-n}{N-1}\right]&\text{ para }k{\neq}l\\ \frac{n}{N^2}\left(N-n\right)&\text{ para }k{=}l \end{cases} \end{align} \]
\[ \begin{align} V_{MAS}[\widehat{t}_{y,\pi}]&=V_{MAS}\left[\frac{N}{n}{\sum}_{s}y_k\right]\\ &=\frac{N^2}{n^2}V_{MAS}\left[{\sum}_{U}I_k(s)y_k\right]\\ &=\frac{N^2}{n^2}V_{MAS}\left[{\sum\sum}_{U}I_k(s)y_k\right]\\ &=\frac{N^2}{n^2}{\sum\sum}_{U}C_{MAS}\left[I_k(s)y_k,I_l(s)y_l\right]\\ &=\frac{N^2}{n^2}\left\{{\sum}_{k=l}V_{MAS}\left[I_k(s)y_k\right]+{\sum\sum}_{k{\neq}l}C_{MAS}\left[I_k(s)y_k,I_l(s)y_l\right]\right\}\\ &=\frac{N^2}{n^2}\left\{{\sum}_{k=l}V_{MAS}\left[I_k(s)\right]y_k^2+{\sum\sum}_{k{\neq}l}C_{MAS}\left[I_k(s),I_l(s)\right]y_ky_l\right\}\\ &=\frac{N^2}{n^2}\left\{{\sum}_{k=l}\frac{n}{N^2}\left(N-n\right)y_k^2-{\sum\sum}_{k{\neq}l}\frac{n}{N^2}\left[\frac{N-n}{N-1}\right]y_ky_l\right\}\\ &=\frac{N^2}{n^2}\left\{\frac{n}{N^2}\left(N-n\right){\sum}_{k=l}y_k^2-\frac{n}{N^2}\left[\frac{N-n}{N-1}\right]{\sum\sum}_{k{\neq}l}y_ky_l\right\}\\ &=\frac{N^2}{n^2}\frac{n}{N^2}\left(N-n\right)\left\{{\sum}_{k=l}y_k^2-\frac{1}{N-1}{\sum\sum}_{k{\neq}l}y_ky_l\right\}\\ &=\frac{1}{n}\left(N-n\right)\left\{{\sum}_{U}y_k^2-\frac{1}{N-1}\left[\left({{\sum}_{U}y_k}\right)^2-{\sum}_{U}y_k^2\right]\right\}\\ &=\frac{1}{n}\left(N-n\right)\frac{1}{N-1}\left\{\left(N-1\right){\sum}_{U}y_k^2-\left[\left({{\sum}_{U}y_k}\right)^2-{\sum}_{U}y_k^2\right]\right\}\\ &=\frac{1}{n}\left(N-n\right)\frac{1}{N-1}\left\{N{\sum}_{U}y_k^2-{\sum}_{k=l}y_k^2-\left({{\sum}_{U}y_k}\right)^2+{\sum}_{U}y_k^2\right\}\\ &=\frac{1}{n}\left(N-n\right)\frac{1}{N-1}\left\{N{\sum}_{U}y_k^2-\left({{\sum}_{U}y_k}\right)^2\right\}\\ &=\frac{N}{n}\left(N-n\right)\frac{1}{N-1}\left\{{\sum}_{U}y_k^2-\frac{1}{N}\left({{\sum}_{U}y_k}\right)^2\right\}\\ &=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\frac{1}{N-1}\left\{{\sum}_{U}y_k^2-{N}\left[\frac{{\sum}_{U}y_k}{N}\right]^2\right\}\\ &=\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{MAS}[\widehat{t}_{y,\pi}]&=\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{ys}^{2} \end{align} \]
\[ \begin{align} E\left[S_{ys}^{2}\right]&=E\left\{\frac{1}{n-1}\left[{\sum}_{s}y_k^2-{n}\bar{y}_{s}^2\right]\right\}\\ &=\frac{1}{n-1}E\left[{\sum}_{s}y_k^2-{n}\bar{y}_{s}^2\right]\\ &=\frac{1}{n-1}\left\{E\left[{\sum}_{s}y_k^2\right]-{n}E\left[\bar{y}_{s}^2\right]\right\}\\ &=\frac{1}{n-1}\left\{E\left[{\sum}_{U}I_k(s)y_k^2\right]-{n}E\left[\frac{\widehat{t}_{y,\pi}^2}{N^2}\right]\right\}\\ &=\frac{1}{n-1}\left\{{\sum}_{U}E\left[I_k(s)\right]y_k^2-\frac{n}{N^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{1}{n-1}\left\{{\sum}_{U}\pi_ky_k^2-\frac{n}{N^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{1}{n-1}\left\{\frac{n}{N}{\sum}_{U}y_k^2-\frac{n}{N^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{n}{n-1}\left\{\frac{1}{N}{\sum}_{U}y_k^2-\frac{1}{N^2}\left\{V_{MAS}\left[\widehat{t}_{y,\pi}^2\right]-{t}_{y,\pi}^{2}\right\}\right\}\\ &=\frac{n}{n-1}\left\{\frac{1}{N}{\sum}_{U}y_k^2-\frac{1}{N^2}\frac{N^2}{n}\left[1-\frac{n}{N}\right]S_{yU}^{2}-\frac{1}{N^2}{t}_{y,\pi}^{2}\right\}\\ &=\frac{n}{n-1}\left\{\frac{1}{N}{\sum}_{U}y_k^2-\frac{1}{n}\left[1-\frac{n}{N}\right]S_{yU}^{2}-\frac{1}{N^2}{t}_{y,\pi}^{2}\right\}\\ &=\frac{n}{n-1}\left\{\frac{1}{N}{\sum}_{U}y_k^2-\frac{1}{N^2}{t}_{y,\pi}^{2}-\frac{1}{n}\left[\frac{N}{N}-\frac{n}{N}\right]S_{yU}^{2}\right\}\\ &=\frac{n}{n-1}\left\{\frac{1}{N}\left[{\sum}_{U}y_k^2-\frac{1}{N}{t}_{y,\pi}^{2}\right]-\frac{1}{n}\left[\frac{N-n}{N}\right]S_{yU}^{2}\right\}\\ &=\frac{n}{n-1}\left\{\frac{N-1}{N}\frac{1}{N-1}\left[{\sum}_{U}y_k^2-{N}\frac{{t}_{y,\pi}^{2}}{N^{2}}\right]-\frac{N-n}{nN}S_{yU}^{2}\right\}\\ &=\frac{n}{n-1}\left\{\frac{N-1}{N}\frac{1}{N-1}\left[{\sum}_{U}y_k^2-{N}\bar{y}_{U}^2\right]-\frac{N-n}{nN}S_{yU}^{2}\right\}\\ &=\frac{n}{n-1}\left\{\frac{N-1}{N}S_{yU}^{2}-\frac{N-n}{nN}S_{yU}^{2}\right\}\\ &=\frac{n}{n-1}\left\{\frac{nN-n}{nN}S_{yU}^{2}-\frac{N-n}{nN}S_{yU}^{2}\right\}\\ &=\frac{n}{n-1}\frac{nN-n-(N-n)}{nN}S_{yU}^{2}\\ &=\frac{n}{n-1}\frac{nN-n-N+n}{nN}S_{yU}^{2}\\ &=\frac{n}{n-1}\frac{nN-N}{nN}S_{yU}^{2}\\ &=\frac{n}{n-1}\frac{n-1}{n}S_{yU}^{2}\\ &=S_{yU}^{2} \end{align} \]
\[ \begin{align} S_{ys}^{2}&=\frac{1}{n-1}\left\{{\sum}_{s}y_k^2-{n}\left[\frac{{\sum}_{s}y_k}{n}\right]^2\right\}\\ &=\frac{1}{n-1}\left[{\sum}_{s}y_k^2-{n}\bar{y}_{s}^2\right]\\ &=\frac{1}{n-1}\left[{\sum}_{s}y_k^2-2{n}\bar{y}_{s}^2+{n}\bar{y}_{s}^2\right]\\ &=\frac{1}{n-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{n}\bar{y}_{s}+{n}\bar{y}_{s}^2\right]\\ &=\frac{1}{n-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{n}\frac{{\sum}_{s}y_k}{n}+{n}\bar{y}_{s}^2\right]\\ &=\frac{1}{n-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{\sum}_{s}y_k+{\sum}_{s}\bar{y}_{s}^2\right]\\ &=\frac{1}{n-1}{\sum}_{s}\left(y_k^2-2y_k\bar{y}_{s}+\bar{y}_{s}^2\right)\\ &=\frac{1}{n-1}{\sum}_{s}\left(y_k-\bar{y}_{s}\right)^2\\ \end{align} \]
\[ \begin{align} \widehat{t}_{y,\pi}(s=U)&={\sum}_{s=U}\frac{y_k}{\pi_k}\\ &=\frac{N}{N}{\sum}_{s=U}y_k\\ &={t}_{y} \end{align} \]
\[ \begin{align} \widehat{t}_{y,alt}&=N\tilde{y}_{s}\\ &=N\frac{{\sum}_{s}\frac{y_k}{{\pi}_{k}}}{{\sum}_{s}\frac{1}{{\pi}_{k}}}\\ &=N\frac{\widehat{t}_{y\pi}}{\widehat{N}}\\ &=N\frac{\widehat{t}_{y\pi}}{n{(s)}}\\ &=N\bar{y}_{s}\\ \end{align} \]
9.2.2.1 Estimador del promedio
\[ \begin{align} \widehat{\bar{y}}_{\pi}&=\frac{\widehat{t}_{y,\pi}}{N}\\ &=\frac{{\sum}_{s}\frac{y_k}{{\pi}_{k}}}{N}\\ &=\frac{\frac{N}{n}{\sum}_{s}{y_k}}{\frac{N}{1}}\\ &=\frac{{\sum}_{s}{y_k}}{n}\\ &=\bar{y}_{s}\\ \end{align} \]
\[ \begin{align} {V}_{MAS}[\widehat{\bar{y}}_{\pi}]&=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\frac{1}{N^2}S_{yU}^{2}\\ &=\frac{1}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{MAS}[\widehat{\bar{y}}_{\pi}]&=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\frac{1}{N^2}S_{ys}^{2}\\ &=\frac{1}{n}\left(1-\frac{n}{N}\right)S_{ys}^{2} \end{align} \]
9.2.2.1.1 Factor de corrección para poblaciones finitas
\[\left(1-\frac{n}{N}\right)\stackrel{N{\rightarrow}\infty}{\implies}{1}\]
9.2.2.1.2 \(n=100\) y \(N=100000\)
\[ \begin{align} {V}_{MAS}[\widehat{\bar{y}}_{\pi}]&=\frac{1}{100}\left(1-\frac{100}{100000}\right)S_{yU}^{2}\\ &=\frac{1}{100}\left(1-\frac{100}{10^{5}}\right)S_{yU}^{2}\\ &=0.01\left(1-0.001\right)S_{yU}^{2}\\ &=0.01\left(0.999\right)S_{yU}^{2} \end{align} \]
9.2.2.1.3 \(n=100\) y \(N=100000000\)
\[ \begin{align} {V}_{MAS}[\widehat{\bar{y}}_{\pi}]&=\frac{1}{100}\left(1-\frac{100}{100000000}\right)S_{yU}^{2}\\ &=\frac{1}{100}\left(1-\frac{100}{10^{8}}\right)S_{yU}^{2}\\ &=0.01\left(1-10^{-6}\right)S_{yU}^{2}\\ &=0.01\left(0.999999\right)S_{yU}^{2} \end{align} \]
9.2.3 Tamaño de muestra
\[ \begin{align} \bar{y}{\pm}Z_{1-\frac{\alpha}{2}}\sqrt{\frac{1}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}&=\bar{y}{\pm}Z_{1-\frac{\alpha}{2}}\sqrt{\frac{1}{n}\left(1-\frac{n}{N}\right)}S_{yU}\\ &=\bar{y}{\pm}Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}} \end{align} \]
\[ \begin{align} \mathcal{P}{\left[|\bar{y}_{S}-\bar{y}_{U}|{\leq}c\right]}&=1-\alpha \end{align} \]
\[ \begin{align} c=Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}}&{\implies}\sqrt{n}=Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{c}\\ &{\implies}n=Z_{1-\frac{\alpha}{2}}^2\left(1-\frac{n}{N}\right)\frac{S_{yU}^2}{c^2}\\ &{\implies}n=\left(Z_{1-\frac{\alpha}{2}}^2-Z_{1-\frac{\alpha}{2}}^2\frac{n}{N}\right)\frac{S_{yU}^2}{c^2}\\ &{\implies}n=Z_{1-\frac{\alpha}{2}}^2\frac{S_{yU}^2}{c^2}-Z_{1-\frac{\alpha}{2}}^2\frac{n}{N}\frac{S_{yU}^2}{c^2}\\ &{\implies}n+Z_{1-\frac{\alpha}{2}}^2\frac{n}{N}\frac{S_{yU}^2}{c^2}=Z_{1-\frac{\alpha}{2}}^2\frac{S_{yU}^2}{c^2}\\ &{\implies}n\left(1+Z_{1-\frac{\alpha}{2}}^2\frac{1}{N}\frac{S_{yU}^2}{c^2}\right)=Z_{1-\frac{\alpha}{2}}^2\frac{S_{yU}^2}{c^2}\\ &{\implies}n=\frac{Z_{1-\frac{\alpha}{2}}^2\frac{S_{yU}^2}{c^2}}{1+Z_{1-\frac{\alpha}{2}}^2\frac{1}{N}\frac{S_{yU}^2}{c^2}}\\ \end{align} \]
\[ \begin{align} \text{argmin}_{n\in\mathbb{Z}^+}Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}}&{\implies}n{\geq}\frac{Z_{1-\frac{\alpha}{2}}^2\frac{S_{yU}^2}{c^2}}{1+Z_{1-\frac{\alpha}{2}}^2\frac{1}{N}\frac{S_{yU}^2}{c^2}}\\ \end{align} \]
\[ \begin{align} \mathcal{P}{\left[|\frac{\bar{y}_{S}-\bar{y}_{U}}{\bar{y}_{U}}|{\leq}c\right]}&=1-\alpha\\ \mathcal{P}{\left[|\bar{y}_{S}-\bar{y}_{U}|{\leq}c|\bar{y}_{U}|\right]}&= \end{align} \]
\[ \begin{align} c|\bar{y}_{U}|=Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}}&{\implies}\sqrt{n}|\bar{y}_{U}|=Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{c}\\ &{\implies}n\bar{y}_{U}^{2}=Z_{1-\frac{\alpha}{2}}^{2}\left(1-\frac{n}{N}\right)\frac{S_{yU}^{2}}{c^{2}}\\ &{\implies}n\bar{y}_{U}^{2}=\left(Z_{1-\frac{\alpha}{2}}^{2}-Z_{1-\frac{\alpha}{2}}^{2}\frac{n}{N}\right)\frac{S_{yU}^{2}}{c^{2}}\\ &{\implies}n\bar{y}_{U}^{2}=Z_{1-\frac{\alpha}{2}}^{2}\frac{S_{yU}^{2}}{c^{2}}-Z_{1-\frac{\alpha}{2}}^{2}\frac{n}{N}\frac{S_{yU}^{2}}{c^{2}}\\ &{\implies}n\bar{y}_{U}^{2}+Z_{1-\frac{\alpha}{2}}^{2}\frac{n}{N}\frac{S_{yU}^{2}}{c^{2}}=Z_{1-\frac{\alpha}{2}}^{2}\frac{S_{yU}^{2}}{c^{2}}\\ &{\implies}n\left(\bar{y}_{U}^{2}+Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{N}\frac{S_{yU}^{2}}{c^{2}}\right)=Z_{1-\frac{\alpha}{2}}^{2}\frac{S_{yU}^{2}}{c^{2}}\\ &{\implies}n=\frac{Z_{1-\frac{\alpha}{2}}^{2}\frac{S_{yU}^{2}}{c^{2}}}{\bar{y}_{U}^{2}+Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{N}\frac{S_{yU}^{2}}{c^{2}}}\\ &{\implies}n=\frac{Z_{1-\frac{\alpha}{2}}^{2}\frac{S_{yU}^{2}}{c^{2}}\frac{1}{\bar{y}_{U}^{2}}}{\bar{y}_{U}^{2}\frac{1}{\bar{y}_{U}^{2}}+Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{N}\frac{S_{yU}^{2}}{c^{2}}\frac{1}{\bar{y}_{U}^{2}}}\\ &{\implies}n=\frac{Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{c^{2}}\frac{S_{yU}^{2}}{\bar{y}_{U}^{2}}}{\frac{\bar{y}_{U}^{2}}{\bar{y}_{U}^{2}}+Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{N}\frac{1}{c^{2}}\frac{S_{yU}^{2}}{\bar{y}_{U}^{2}}}\\ &{\implies}n=\frac{Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{c^{2}}{CV}_{yU}^{2}}{{1}+Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{N}\frac{1}{c^{2}}{CV}_{yU}^{2}}\\ &{\implies}n=\frac{Z_{1-\frac{\alpha}{2}}^{2}\frac{{CV}_{yU}^{2}}{c^{2}}}{{1}+Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{N}\frac{{CV}_{yU}^{2}}{c^{2}}}\\ \end{align} \]
\[ \begin{align} \text{argmin}_{n\in\mathbb{Z}^+}Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}}&{\implies}n{\geq}\frac{Z_{1-\frac{\alpha}{2}}^{2}\frac{{CV}_{yU}^{2}}{c^{2}}}{{1}+Z_{1-\frac{\alpha}{2}}^{2}\frac{1}{N}\frac{{CV}_{yU}^{2}}{c^{2}}}\\ \end{align} \]
9.2.3.1 ¿Bajo qué condiciones?
Condiciones de regularidad de Noether
\(N_{\nu}-n_{\nu}{\rightarrow}+{\infty}\)
\(y_{\nu_{i}}\), \(i=1,2,\ldots,N_{\nu}\)
\(|y_{\nu_{i}}-\overline{y}_{\nu}|>\tau\sqrt{n_{\nu}(1-f_{\nu})}S_{\nu}\)
\[{\lim}_{\nu{\rightarrow}+{\infty}}\frac{{\sum}_{S_{\nu}}\left({y_{\nu_{i}}-\overline{y}_{\nu}}\right)^{2}}{\left(N_{\nu}-1\right)S_{\nu}^{2}}{\sim}N{(0,1)}\]
- \(n{\rightarrow}{\infty}\text{, }N{\rightarrow}{\infty}\text{ y }N-n{\rightarrow}{\infty}{\implies}\frac{\bar{y}_{S}-\bar{y}_{U}}{\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}}}{\sim}N{(0,1)}\)
“Por supuesto, algunas cantidades poblacionales necesarias para estimar el tamaño de la muestra no se conocen; de hecho, si se conocieran, no habría necesidad de realizar estudio alguno, porque directamente se concernirían los parametros poblacionales de interés” Andres Gutíerrez (2009)
Lohr (2000) considera los siguientes escenarios para realizar una estimación previa de los parámetros de interés
Realizar una prueba piloto, unas cuantas entrevistas conforman la muestra piloto, seleccionada con el mismo diseño de muestreo genérico.
Utilizar información a priorí de estudios anteriores.
“Estime la varianza ajustando una distribución teórica a la característica de interés” Ospina (2001), “cuando su desconocimineto sea absoluto use una distribución uniforme” Wu (2003)
9.2.3.2 Estimación del tamaño de muestra y total
N <- dim(Lucy)[1];N## [1] 2396n <- 30; n## [1] 30seleccion <- sample(N,n)
muestra <- Lucy[seleccion,]SyU2 <- var(muestra$Income); SyU2## [1] 38213.93ybar <- mean(muestra$Income); ybar## [1] 391.0667c <- 19; c## [1] 19\[ \begin{align} \text{argmin}_{n\in\mathbb{Z}^+}Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}}{\implies}n{\geq}\frac{Z_{1-\frac{\alpha}{2}}^2\frac{S_{yU}^2}{c^2}}{1+Z_{1-\frac{\alpha}{2}}^2\frac{1}{N}\frac{S_{yU}^2}{c^2}}&=\frac{1.96^2\frac{3.821393\times 10^{4}}{c^2}}{1+1.96^2\frac{1}{2396}\frac{3.821393\times 10^{4}}{c^2}}\\ &=\frac{1.96^2\frac{3.821393\times 10^{4}}{361}}{1+1.96^2\frac{1}{2396}\frac{3.821393\times 10^{4}}{361}}\\ &=\frac{3.84\frac{3.821393\times 10^{4}}{361}}{1+3.84\frac{1}{2396}\frac{3.821393\times 10^{4}}{361}}\\ &=\frac{3.84*105.8558}{1+3.84*4\times 10^{-4}*105.8558}\\ &=\frac{406.6405}{1+0.1697164}\\ &=348 \end{align} \]
9.2.3.2.1 \(n=348\)
attach(Lucy)seleccion <- S.SI(N,n)
muestra <- Lucy[seleccion,]
attach(muestra);muestra## ID Ubication Level Zone Income Employees Taxes SPAM
## 2 AB002 c1k2 Small A 329 19 4.0 yes
## 12 AB012 c1k12 Small A 419 20 7.0 no
## 22 AB022 c1k22 Small A 381 42 6.0 yes
## 33 AB033 c1k33 Small A 334 72 5.0 yes
## 58 AB061 c1k58 Small A 363 73 6.0 no
## 72 AB077 c1k72 Small A 363 81 5.0 yes
## 81 AB086 c1k81 Small A 330 95 4.0 no
## 86 AB099 c1k86 Small A 456 75 9.0 yes
## 88 AB101 c1k88 Small A 494 34 10.5 no
## 90 AB106 c1k90 Small A 450 87 9.0 yes
## 95 AB1151 c1k95 Small A 490 98 10.5 yes
## 98 AB1193 c1k98 Small B 490 38 10.5 yes
## 99 AB1194 c1k99 Small B 480 49 10.0 no
## 105 AB1229 c2k6 Small B 388 91 6.0 no
## 112 AB126 c2k13 Small B 65 69 0.5 yes
## 120 AB132 c2k21 Small B 125 34 0.5 no
## 131 AB1340 c2k32 Small B 222 34 2.0 yes
## 132 AB1341 c2k33 Small B 304 18 4.0 no
## 142 AB1350 c2k43 Small B 280 69 3.0 no
## 150 AB1358 c2k51 Small B 384 70 6.0 no
## 160 AB1367 c2k61 Small B 200 49 1.0 yes
## 167 AB1373 c2k68 Small B 355 33 5.0 yes
## 171 AB1377 c2k72 Small B 280 65 3.0 yes
## 174 AB138 c2k75 Small B 70 17 0.5 no
## 190 AB1394 c2k91 Small B 296 69 3.0 no
## 198 AB1401 c2k99 Small B 201 81 1.0 yes
## 220 AB1421 c3k22 Small B 370 30 6.0 no
## 231 AB1431 c3k33 Small B 240 79 2.0 no
## 236 AB1436 c3k38 Small B 215 86 2.0 no
## 240 AB144 c3k42 Small B 103 81 0.5 no
## 265 AB1462 c3k67 Small B 330 32 4.0 yes
## 268 AB1465 c3k70 Small B 258 84 2.0 no
## 273 AB147 c3k75 Small B 144 63 0.5 yes
## 276 AB1472 c3k78 Small B 373 90 6.0 yes
## 282 AB1478 c3k84 Small B 267 88 3.0 no
## 285 AB1480 c3k87 Small B 210 14 1.0 yes
## 291 AB1486 c3k93 Small B 320 51 4.0 no
## 303 AB1497 c4k6 Small B 330 27 4.0 yes
## 306 AB150 c4k9 Small B 102 81 0.5 no
## 311 AB1504 c4k14 Small B 316 19 4.0 yes
## 313 AB1506 c4k16 Small B 275 68 3.0 yes
## 314 AB1507 c4k17 Small B 230 38 2.0 yes
## 317 AB151 c4k20 Small B 159 76 1.0 yes
## 324 AB1516 c4k27 Small B 260 32 2.0 yes
## 327 AB1519 c4k30 Small B 310 70 4.0 yes
## 328 AB152 c4k31 Small B 193 81 1.0 no
## 335 AB1526 c4k38 Small B 218 70 2.0 yes
## 337 AB1528 c4k40 Small B 235 51 2.0 no
## 371 AB1559 c4k74 Small B 260 76 2.0 no
## 376 AB1563 c4k79 Small B 380 94 6.0 no
## 387 AB1573 c4k90 Small B 248 83 2.0 yes
## 388 AB1574 c4k91 Small B 230 86 2.0 no
## 392 AB1578 c4k95 Small B 315 31 4.0 no
## 395 AB1580 c4k98 Small B 299 85 3.0 no
## 398 AB1583 c5k2 Small B 277 45 3.0 yes
## 400 AB1585 c5k4 Small B 300 54 3.0 yes
## 411 AB1595 c5k15 Small B 343 60 5.0 no
## 416 AB160 c5k20 Small B 179 16 1.0 no
## 427 AB161 c5k31 Small B 162 24 1.0 yes
## 428 AB1610 c5k32 Small B 291 85 3.0 no
## 443 AB1624 c5k47 Small B 291 45 3.0 yes
## 445 AB1626 c5k49 Small B 331 16 4.0 no
## 452 AB1632 c5k56 Small B 281 85 3.0 no
## 453 AB1633 c5k57 Small B 338 96 5.0 yes
## 455 AB1635 c5k59 Small B 338 52 5.0 yes
## 459 AB1639 c5k63 Small B 331 40 4.0 yes
## 468 AB1647 c5k72 Small B 354 49 5.0 no
## 474 AB1652 c5k78 Small B 369 97 6.0 yes
## 478 AB1656 c5k82 Small B 233 35 2.0 no
## 485 AB1662 c5k89 Small B 243 27 2.0 yes
## 488 AB1665 c5k92 Small B 265 48 3.0 no
## 491 AB1668 c5k95 Small B 336 80 5.0 yes
## 503 AB1679 c6k8 Small B 250 59 2.0 yes
## 509 AB1684 c6k14 Small B 340 44 5.0 no
## 510 AB1685 c6k15 Small B 235 15 2.0 no
## 531 AB1704 c6k36 Small B 189 25 1.0 yes
## 538 AB1710 c6k43 Small B 1 45 0.5 yes
## 539 AB1711 c6k44 Small B 180 12 1.0 yes
## 540 AB1712 c6k45 Small B 172 48 1.0 yes
## 544 AB1716 c6k49 Small B 125 74 0.5 no
## 548 AB172 c6k53 Small B 80 77 0.5 yes
## 558 AB1729 c6k63 Small B 106 53 0.5 yes
## 564 AB1734 c6k69 Small B 137 23 0.5 no
## 572 AB1741 c6k77 Small B 143 7 0.5 yes
## 581 AB175 c6k86 Small B 71 69 0.5 no
## 583 AB1751 c6k88 Small B 170 36 1.0 yes
## 590 AB1758 c6k95 Small B 152 43 1.0 no
## 597 AB1764 c7k3 Small B 189 53 1.0 no
## 598 AB1765 c7k4 Small B 226 78 2.0 no
## 616 AB1781 c7k22 Small B 188 77 1.0 yes
## 624 AB1789 c7k30 Small B 207 30 1.0 no
## 635 AB1799 c7k41 Small B 130 38 0.5 yes
## 645 AB1808 c7k51 Small B 104 41 0.5 no
## 665 AB1826 c7k71 Small B 180 68 1.0 yes
## 667 AB1828 c7k73 Small B 94 41 0.5 no
## 681 AB1840 c7k87 Small B 185 64 1.0 yes
## 688 AB1847 c7k94 Small B 181 8 1.0 yes
## 689 AB1848 c7k95 Small B 179 24 1.0 yes
## 691 AB185 c7k97 Small C 68 73 0.5 yes
## 696 AB1854 c8k3 Small C 211 82 1.0 yes
## 707 AB1864 c8k14 Small C 189 9 1.0 yes
## 710 AB1867 c8k17 Small C 197 33 1.0 yes
## 711 AB1868 c8k18 Small C 183 12 1.0 no
## 715 AB1871 c8k22 Small C 160 32 1.0 yes
## 718 AB1874 c8k25 Small C 182 40 1.0 yes
## 724 AB188 c8k31 Small C 117 82 0.5 yes
## 738 AB1892 c8k45 Small C 490 30 10.5 yes
## 741 AB1895 c8k48 Small C 350 48 5.0 yes
## 742 AB1896 c8k49 Small C 319 67 4.0 yes
## 744 AB1898 c8k51 Small C 370 98 6.0 yes
## 750 AB1903 c8k57 Small C 420 32 8.0 no
## 752 AB1906 c8k59 Small C 460 43 9.0 no
## 764 AB1918 c8k71 Small C 390 39 7.0 yes
## 766 AB192 c8k73 Small C 198 13 1.0 no
## 792 AB1947 c8k99 Small C 385 54 6.0 no
## 801 AB1955 c9k9 Small C 440 82 8.0 no
## 802 AB1956 c9k10 Small C 470 76 10.0 yes
## 806 AB1960 c9k14 Small C 312 74 4.0 yes
## 807 AB1961 c9k15 Small C 430 21 8.0 no
## 809 AB1963 c9k17 Small C 400 95 7.0 no
## 821 AB1974 c9k29 Small C 410 28 7.0 yes
## 825 AB1978 c9k33 Small C 471 101 10.0 no
## 830 AB1982 c9k38 Small C 440 74 8.0 yes
## 831 AB1983 c9k39 Small C 440 94 8.0 yes
## 832 AB1984 c9k40 Small C 480 25 10.0 yes
## 838 AB1990 c9k46 Small C 378 62 6.0 yes
## 840 AB1992 c9k48 Small C 430 37 8.0 no
## 844 AB1996 c9k52 Small C 400 23 7.0 no
## 846 AB1998 c9k54 Small C 380 18 6.0 no
## 856 AB2008 c9k64 Small C 434 45 8.0 yes
## 861 AB2016 c9k69 Small C 436 97 8.0 yes
## 877 AB2034 c9k85 Small C 434 53 8.0 yes
## 881 AB2038 c9k89 Small C 460 99 9.0 yes
## 888 AB2045 c9k96 Small C 440 34 8.0 yes
## 903 AB2064 c10k12 Small C 420 76 8.0 yes
## 934 AB2099 c10k43 Small C 463 28 9.0 yes
## 942 AB2107 c10k51 Small C 469 40 10.0 no
## 949 AB2116 c10k58 Small C 457 95 9.0 no
## 951 AB2118 c10k60 Small C 480 85 10.0 no
## 956 AB2122 c10k65 Small C 420 28 7.0 no
## 963 AB2130 c10k72 Small C 460 67 9.0 no
## 964 AB2131 c10k73 Small C 334 76 5.0 yes
## 965 AB2132 c10k74 Small C 380 22 6.0 no
## 966 AB2133 c10k75 Small C 370 66 6.0 no
## 967 AB2134 c10k76 Small C 360 29 5.0 yes
## 968 AB2135 c10k77 Small C 450 90 9.0 yes
## 977 AB2145 c10k86 Small C 480 81 10.0 yes
## 982 AB215 c10k91 Small C 180 36 1.0 yes
## 985 AB2152 c10k94 Small C 452 63 9.0 no
## 1010 AB2178 c11k20 Small C 494 38 10.5 no
## 1018 AB2188 c11k28 Small C 386 23 6.0 yes
## 1023 AB2194 c11k33 Small C 479 69 10.0 no
## 1024 AB2195 c11k34 Small C 445 38 9.0 yes
## 1026 AB2198 c11k36 Small C 366 69 6.0 no
## 1028 AB220 c11k38 Small C 182 24 1.0 yes
## 1032 AB2203 c11k42 Small C 421 85 8.0 yes
## 1035 AB2207 c11k45 Small C 350 24 5.0 no
## 1042 AB2213 c11k52 Small C 400 47 7.0 yes
## 1047 AB2219 c11k57 Small C 350 25 5.0 yes
## 1057 AB2230 c11k67 Small C 343 80 5.0 no
## 1064 AB2237 c11k74 Small C 487 53 10.5 no
## 1066 AB224 c11k76 Small C 143 43 0.5 yes
## 1079 AB2252 c11k89 Small C 341 24 5.0 yes
## 1080 AB2254 c11k90 Small C 470 40 10.0 no
## 1084 AB2259 c11k94 Small C 463 24 10.0 yes
## 1086 AB2260 c11k96 Small C 443 30 8.0 no
## 1105 AB2280 c12k16 Small C 430 61 8.0 yes
## 1110 AB2286 c12k21 Small C 400 19 7.0 no
## 1127 AB2301 c12k38 Small C 457 67 9.0 no
## 1129 AB2303 c12k40 Small C 468 32 10.0 yes
## 1132 AB2309 c12k43 Small C 495 38 10.5 no
## 1136 AB2312 c12k47 Small C 424 61 8.0 yes
## 1146 AB2321 c12k57 Small C 454 51 9.0 yes
## 1148 AB2323 c12k59 Small C 464 32 10.0 yes
## 1149 AB2324 c12k60 Small C 356 61 5.0 yes
## 1162 AB2339 c12k73 Small C 425 21 8.0 yes
## 1173 AB235 c12k84 Small C 65 37 0.5 yes
## 1176 AB2353 c12k87 Small C 484 25 10.5 no
## 1181 AB2358 c12k92 Small C 348 56 5.0 yes
## 1194 AB237 c13k6 Small C 166 64 1.0 no
## 1200 AB2375 c13k12 Small C 489 101 10.5 yes
## 1206 AB2384 c13k18 Small C 361 65 5.0 yes
## 1229 AB250 c13k41 Small C 98 9 0.5 no
## 1242 AB263 c13k54 Small C 177 48 1.0 no
## 1252 AB273 c13k64 Small C 183 84 1.0 yes
## 1253 AB274 c13k65 Small C 117 62 0.5 yes
## 1255 AB276 c13k67 Small C 119 38 0.5 yes
## 1256 AB277 c13k68 Small C 130 46 0.5 yes
## 1272 AB293 c13k84 Small C 77 69 0.5 no
## 1273 AB294 c13k85 Small C 103 25 0.5 no
## 1287 AB308 c13k99 Small C 120 46 0.5 yes
## 1290 AB311 c14k3 Small C 122 42 0.5 yes
## 1300 AB321 c14k13 Small C 133 35 0.5 yes
## 1302 AB323 c14k15 Small C 151 75 1.0 no
## 1311 AB332 c14k24 Small C 172 24 1.0 yes
## 1315 AB336 c14k28 Small C 131 22 0.5 no
## 1316 AB337 c14k29 Small C 131 74 0.5 no
## 1317 AB338 c14k30 Small C 127 46 0.5 yes
## 1318 AB339 c14k31 Small C 122 30 0.5 yes
## 1320 AB341 c14k33 Small C 151 59 1.0 no
## 1325 AB346 c14k38 Small C 117 62 0.5 yes
## 1330 AB351 c14k43 Small C 135 75 0.5 yes
## 1335 AB356 c14k48 Small C 209 90 1.0 no
## 1336 AB357 c14k49 Small C 200 45 1.0 no
## 1352 AB373 c14k65 Small C 99 21 0.5 yes
## 1355 AB376 c14k68 Small C 109 42 0.5 yes
## 1356 AB377 c14k69 Small C 121 50 0.5 yes
## 1367 AB388 c14k80 Small C 96 29 0.5 yes
## 1371 AB392 c14k84 Small C 124 62 0.5 yes
## 1385 AB406 c14k98 Small C 119 54 0.5 no
## 1388 AB409 c15k2 Small C 131 18 0.5 yes
## 1389 AB410 c15k3 Small C 165 76 1.0 yes
## 1390 AB411 c15k4 Small C 168 32 1.0 yes
## 1406 AB427 c15k20 Small C 206 66 1.0 yes
## 1431 AB452 c15k45 Small C 186 60 1.0 yes
## 1436 AB457 c15k50 Small C 260 80 2.0 yes
## 1444 AB465 c15k58 Small C 233 67 2.0 yes
## 1453 AB474 c15k67 Small C 251 51 2.0 yes
## 1467 AB488 c15k81 Small C 195 61 1.0 no
## 1474 AB495 c15k88 Small C 324 75 4.0 yes
## 1478 AB499 c15k92 Small C 367 85 6.0 yes
## 1479 AB500 c15k93 Small C 278 17 3.0 no
## 1481 AB502 c15k95 Small C 268 52 3.0 no
## 1485 AB506 c15k99 Small C 258 52 2.0 yes
## 1489 AB510 c16k4 Small C 213 50 1.0 yes
## 1490 AB511 c16k5 Small C 236 79 2.0 yes
## 1492 AB513 c16k7 Small C 188 29 1.0 yes
## 1498 AB519 c16k13 Small C 280 21 3.0 yes
## 1521 AB542 c16k36 Small C 364 97 6.0 no
## 1527 AB548 c16k42 Small C 283 77 3.0 no
## 1534 AB555 c16k49 Small C 279 85 3.0 yes
## 1539 AB560 c16k54 Small C 243 75 2.0 yes
## 1543 AB564 c16k58 Small C 302 30 4.0 no
## 1544 AB565 c16k59 Small C 262 16 2.0 yes
## 1545 AB566 c16k60 Small C 200 69 1.0 yes
## 1557 AB649 c16k72 Small D 470 24 10.0 no
## 1558 AB661 c16k73 Small D 460 96 9.0 no
## 1571 AB827 c16k86 Small D 470 84 10.0 yes
## 1575 AB958 c16k90 Small E 490 38 10.5 yes
## 1590 AB1002 c17k6 Medium A 743 127 27.0 no
## 1592 AB1004 c17k8 Medium A 844 141 35.0 no
## 1603 AB1016 c17k19 Medium A 935 119 42.0 no
## 1616 AB1030 c17k32 Medium A 710 92 26.0 yes
## 1620 AB1034 c17k36 Medium A 790 54 31.0 yes
## 1628 AB1047 c17k44 Medium A 601 32 18.0 no
## 1632 AB1050 c17k48 Medium A 550 75 14.0 yes
## 1633 AB1051 c17k49 Medium A 675 48 23.0 no
## 1634 AB1052 c17k50 Medium A 730 117 27.0 no
## 1640 AB1058 c17k56 Medium A 630 95 19.0 yes
## 1645 AB1063 c17k61 Medium A 680 96 23.0 yes
## 1652 AB107 c17k68 Medium A 560 72 15.0 yes
## 1664 AB1081 c17k80 Medium A 814 106 33.0 no
## 1676 AB1095 c17k92 Medium A 986 124 46.0 yes
## 1678 AB1099 c17k94 Medium A 580 97 16.0 yes
## 1685 AB1105 c18k2 Medium A 680 80 23.0 yes
## 1727 AB1157 c18k44 Medium A 794 67 31.0 no
## 1732 AB1162 c18k49 Medium A 660 74 22.0 yes
## 1736 AB1166 c18k53 Medium A 880 60 38.0 yes
## 1747 AB1178 c18k64 Medium A 668 63 22.0 yes
## 1751 AB1182 c18k68 Medium A 840 76 35.0 yes
## 1756 AB1188 c18k73 Medium A 621 59 19.0 yes
## 1785 AB1227 c19k3 Medium B 590 79 16.0 yes
## 1788 AB1233 c19k6 Medium B 600 84 17.0 no
## 1789 AB1234 c19k7 Medium B 704 47 25.0 yes
## 1795 AB1242 c19k13 Medium B 520 85 12.0 yes
## 1797 AB1244 c19k15 Medium B 590 47 17.0 yes
## 1801 AB1248 c19k19 Medium B 519 92 12.0 yes
## 1808 AB1257 c19k26 Medium B 750 88 28.0 yes
## 1809 AB1258 c19k27 Medium B 545 42 14.0 no
## 1817 AB1266 c19k35 Medium B 671 111 23.0 no
## 1829 AB1278 c19k47 Medium B 804 103 32.0 no
## 1831 AB1280 c19k49 Medium B 963 106 45.0 yes
## 1836 AB1286 c19k54 Medium B 890 60 39.0 yes
## 1854 AB1306 c19k72 Medium B 570 101 15.0 yes
## 1860 AB1313 c19k78 Medium B 697 65 24.0 yes
## 1862 AB1315 c19k80 Medium B 915 129 41.0 yes
## 1866 AB1320 c19k84 Medium B 721 65 27.0 yes
## 1884 AB1959 c20k3 Medium C 540 38 13.0 no
## 1896 AB2046 c20k15 Medium C 530 69 13.0 no
## 1913 AB2125 c20k32 Medium C 505 79 12.0 no
## 1920 AB2179 c20k39 Medium C 579 49 16.0 yes
## 1924 AB2192 c20k43 Medium C 530 41 13.0 no
## 1925 AB2197 c20k44 Medium C 550 47 14.0 no
## 1935 AB2263 c20k54 Medium C 523 89 13.0 yes
## 1945 AB2340 c20k64 Medium C 604 81 18.0 yes
## 1952 AB570 c20k71 Medium C 530 49 13.0 no
## 1959 AB577 c20k78 Medium C 620 42 19.0 yes
## 1961 AB579 c20k80 Medium C 700 97 24.0 yes
## 1965 AB583 c20k84 Medium C 665 58 22.0 no
## 1978 AB598 c20k97 Medium C 630 111 20.0 yes
## 1983 AB605 c22k3 Medium C 619 62 19.0 yes
## 2002 AB626 c22k22 Medium D 580 109 16.0 no
## 2008 AB632 c22k28 Medium D 710 103 25.0 no
## 2012 AB638 c22k32 Medium D 570 41 15.0 yes
## 2013 AB642 c22k33 Medium D 570 37 15.0 yes
## 2014 AB643 c22k34 Medium D 540 106 13.0 no
## 2021 AB651 c22k41 Medium D 580 46 16.0 no
## 2023 AB653 c22k43 Medium D 530 73 13.0 yes
## 2040 AB673 c22k60 Medium D 630 59 20.0 yes
## 2043 AB676 c22k63 Medium D 520 101 12.0 yes
## 2048 AB683 c22k68 Medium D 656 90 22.0 no
## 2050 AB685 c22k70 Medium D 676 72 23.0 no
## 2070 AB706 c22k90 Medium D 540 90 13.0 yes
## 2079 AB715 c22k99 Medium D 710 87 25.0 no
## 2085 AB721 c23k6 Medium D 740 94 27.0 yes
## 2087 AB723 c23k8 Medium D 900 97 40.0 yes
## 2092 AB732 c23k13 Medium D 570 73 15.0 no
## 2093 AB733 c23k14 Medium D 680 84 23.0 no
## 2098 AB738 c23k19 Medium D 760 124 29.0 yes
## 2105 AB755 c23k26 Medium D 500 71 11.0 no
## 2126 AB779 c23k47 Medium D 580 50 16.0 yes
## 2131 AB784 c23k52 Medium D 700 70 24.0 yes
## 2160 AB817 c23k81 Medium D 612 58 19.0 yes
## 2171 AB829 c23k92 Medium D 595 87 17.0 yes
## 2184 AB843 c24k6 Medium E 710 91 25.0 no
## 2187 AB846 c24k9 Medium E 750 51 28.0 no
## 2201 AB861 c24k23 Medium E 870 135 38.0 yes
## 2204 AB867 c24k26 Medium E 546 34 14.0 yes
## 2211 AB874 c24k33 Medium E 700 122 24.0 no
## 2214 AB877 c24k36 Medium E 879 103 38.0 yes
## 2219 AB884 c24k41 Medium E 540 82 13.0 yes
## 2227 AB895 c24k49 Medium E 594 99 17.0 yes
## 2228 AB896 c24k50 Medium E 620 98 19.0 yes
## 2229 AB897 c24k51 Medium E 720 100 26.0 yes
## 2234 AB902 c24k56 Medium E 630 91 20.0 yes
## 2240 AB913 c24k62 Medium E 727 73 27.0 yes
## 2254 AB928 c24k76 Medium E 841 137 35.0 no
## 2262 AB937 c24k84 Medium E 519 92 12.0 yes
## 2264 AB939 c24k86 Medium E 634 36 20.0 no
## 2269 AB944 c24k91 Medium E 670 95 23.0 yes
## 2285 AB965 c25k8 Medium E 601 65 18.0 no
## 2293 AB973 c25k16 Medium E 620 86 19.0 yes
## 2296 AB976 c25k19 Medium E 910 109 41.0 yes
## 2302 AB982 c25k25 Medium E 864 138 37.0 no
## 2321 AB1039 c25k44 Big A 1016 96 50.0 no
## 2324 AB1042 c25k47 Big A 1100 161 57.0 yes
## 2327 AB1093 c25k50 Big A 1063 146 54.0 yes
## 2343 AB1187 c25k66 Big A 1459 176 99.0 no
## 2349 AB1217 c25k72 Big B 1101 117 58.0 yes
## 2352 AB1232 c25k75 Big B 1490 201 105.0 yes
## 2358 AB636 c25k81 Big D 1024 149 50.0 yes
## 2359 AB637 c25k82 Big D 1405 111 83.0 yes
## 2363 AB741 c25k86 Big D 1093 119 56.0 no
## 2365 AB744 c25k88 Big D 1370 182 77.0 no
## 2367 AB746 c25k90 Big D 1090 105 55.0 yes
## 2368 AB747 c25k91 Big D 1280 145 65.0 yes
## 2379 AB892 c26k3 Big E 1084 92 54.0 yes
## 2381 AB904 c26k5 Big E 1035 86 51.0 yesa.estimar <- data.frame(Income, Employees, Taxes);E.SI(N,n,a.estimar)## N Income Employees Taxes
## Estimation 2396 1.025082e+06 1.538741e+05 27739.896552
## Standard Error 0 3.116160e+04 3.830948e+03 1800.464032
## CVE 0 3.039914e+00 2.489663e+00 6.490522
## DEFF NaN 1.000000e+00 1.000000e+00 1.000000teorica <- rgamma(N,shape=2.7,scale=180)
CVyU <- sd(teorica)/mean(teorica); CVyU## [1] 0.6080588plot(density(teorica), main="Histograma de los ingresos", xlab="Ingresos", ylab="frecuencia")
hist(Lucy$Income, col=rainbow(7), freq=F, add=T)
c <- 0.038; c## [1] 0.038\[ \begin{align} \text{argmin}_{n\in\mathbb{Z}^+}Z_{1-\frac{\alpha}{2}}\sqrt{\left(1-\frac{n}{N}\right)}\frac{S_{yU}}{\sqrt{n}}{\implies}n{\geq}\frac{Z_{1-\frac{\alpha}{2}}^2\frac{S_{yU}^2}{c^2}}{1+Z_{1-\frac{\alpha}{2}}^2\frac{1}{N}\frac{S_{yU}^2}{c^2}}&=\frac{1.96^2\frac{0.37}{c^2}}{1+1.96^2\frac{1}{2396}\frac{0.37}{c^2}}\\ &=\frac{1.96^2\frac{0.37}{0.001444}}{1+1.96^2\frac{1}{2396}\frac{0.37}{0.001444}}\\ &=\frac{3.84\frac{0.37}{0.001444}}{1+3.84\frac{1}{2396}\frac{0.37}{0.001444}}\\ &=\frac{3.84*256.0496}{1+3.84*4\times 10^{-4}*256.0496}\\ &=\frac{983.6038}{1+0.4105191}\\ &=697 \end{align} \]
9.2.3.2.2 \(n=697\)
attach(Lucy)seleccion <- S.SI(N,n)
muestra <- Lucy[seleccion,]
attach(muestra);muestra## ID Ubication Level Zone Income Employees Taxes SPAM
## 2 AB002 c1k2 Small A 329 19 4.0 yes
## 8 AB008 c1k8 Small A 473 57 10.0 yes
## 10 AB010 c1k10 Small A 361 25 5.0 no
## 12 AB012 c1k12 Small A 419 20 7.0 no
## 14 AB014 c1k14 Small A 330 23 4.0 yes
## 21 AB021 c1k21 Small A 425 49 8.0 yes
## 25 AB025 c1k25 Small A 365 49 6.0 yes
## 27 AB027 c1k27 Small A 400 95 7.0 yes
## 30 AB030 c1k30 Small A 354 33 5.0 yes
## 32 AB032 c1k32 Small A 380 18 6.0 yes
## 33 AB033 c1k33 Small A 334 72 5.0 yes
## 40 AB040 c1k40 Small A 491 86 10.5 yes
## 42 AB042 c1k42 Small A 444 34 8.0 yes
## 44 AB044 c1k44 Small A 337 44 5.0 no
## 48 AB048 c1k48 Small A 422 101 8.0 yes
## 53 AB055 c1k53 Small A 348 20 5.0 yes
## 58 AB061 c1k58 Small A 363 73 6.0 no
## 64 AB067 c1k64 Small A 475 57 10.0 no
## 65 AB068 c1k65 Small A 360 61 5.0 no
## 66 AB069 c1k66 Small A 392 75 7.0 no
## 68 AB071 c1k68 Small A 360 29 5.0 yes
## 69 AB072 c1k69 Small A 390 95 7.0 yes
## 73 AB078 c1k73 Small A 330 79 4.0 yes
## 81 AB086 c1k81 Small A 330 95 4.0 no
## 87 AB100 c1k87 Small A 490 62 10.5 yes
## 88 AB101 c1k88 Small A 494 34 10.5 no
## 91 AB1098 c1k91 Small A 285 65 3.0 no
## 93 AB113 c1k93 Small A 441 66 8.0 no
## 95 AB1151 c1k95 Small A 490 98 10.5 yes
## 97 AB117 c1k97 Small A 378 30 6.0 no
## 102 AB122 c2k3 Small B 66 69 0.5 yes
## 110 AB125 c2k11 Small B 211 26 1.0 yes
## 117 AB130 c2k18 Small B 160 56 1.0 yes
## 119 AB131 c2k20 Small B 84 81 0.5 yes
## 120 AB132 c2k21 Small B 125 34 0.5 no
## 121 AB133 c2k22 Small B 28 73 0.5 yes
## 123 AB1333 c2k24 Small B 312 82 4.0 no
## 124 AB1334 c2k25 Small B 230 42 2.0 yes
## 125 AB1335 c2k26 Small B 290 29 3.0 yes
## 126 AB1336 c2k27 Small B 231 79 2.0 no
## 129 AB1339 c2k30 Small B 320 71 4.0 yes
## 130 AB134 c2k31 Small B 202 37 1.0 yes
## 131 AB1340 c2k32 Small B 222 34 2.0 yes
## 144 AB1352 c2k45 Small B 273 16 3.0 yes
## 145 AB1353 c2k46 Small B 240 83 2.0 yes
## 147 AB1355 c2k48 Small B 266 16 3.0 no
## 156 AB1363 c2k57 Small B 320 11 4.0 no
## 159 AB1366 c2k60 Small B 230 10 2.0 yes
## 163 AB137 c2k64 Small B 109 50 0.5 yes
## 164 AB1370 c2k65 Small B 310 58 4.0 yes
## 166 AB1372 c2k67 Small B 318 51 4.0 yes
## 167 AB1373 c2k68 Small B 355 33 5.0 yes
## 168 AB1374 c2k69 Small B 314 78 4.0 yes
## 169 AB1375 c2k70 Small B 265 48 3.0 yes
## 171 AB1377 c2k72 Small B 280 65 3.0 yes
## 180 AB1385 c2k81 Small B 340 60 5.0 yes
## 182 AB1387 c2k83 Small B 260 88 2.0 yes
## 189 AB1393 c2k90 Small B 260 32 2.0 yes
## 190 AB1394 c2k91 Small B 296 69 3.0 no
## 193 AB1397 c2k94 Small B 350 48 5.0 yes
## 194 AB1398 c2k95 Small B 330 39 4.0 yes
## 195 AB1399 c2k96 Small B 230 23 2.0 yes
## 198 AB1401 c2k99 Small B 201 81 1.0 yes
## 199 AB1402 c3k1 Small B 280 61 3.0 yes
## 210 AB1412 c3k12 Small B 314 58 4.0 yes
## 213 AB1415 c3k15 Small B 240 87 2.0 yes
## 214 AB1416 c3k16 Small B 300 18 3.0 yes
## 220 AB1421 c3k22 Small B 370 30 6.0 no
## 225 AB1426 c3k27 Small B 330 67 4.0 yes
## 228 AB1429 c3k30 Small B 370 58 6.0 no
## 232 AB1432 c3k34 Small B 240 87 2.0 yes
## 233 AB1433 c3k35 Small B 211 26 1.0 yes
## 234 AB1434 c3k36 Small B 257 44 2.0 yes
## 239 AB1439 c3k41 Small B 297 57 3.0 no
## 240 AB144 c3k42 Small B 103 81 0.5 no
## 242 AB1441 c3k44 Small B 351 29 5.0 yes
## 245 AB1444 c3k47 Small B 390 19 7.0 yes
## 246 AB1445 c3k48 Small B 312 86 4.0 yes
## 249 AB1448 c3k51 Small B 301 14 4.0 yes
## 251 AB145 c3k53 Small B 62 57 0.5 no
## 258 AB1456 c3k60 Small B 225 34 2.0 no
## 265 AB1462 c3k67 Small B 330 32 4.0 yes
## 275 AB1471 c3k77 Small B 281 25 3.0 yes
## 279 AB1475 c3k81 Small B 210 10 1.0 yes
## 280 AB1476 c3k82 Small B 260 84 2.0 yes
## 281 AB1477 c3k83 Small B 307 26 4.0 yes
## 284 AB148 c3k86 Small B 91 29 0.5 no
## 287 AB1482 c3k89 Small B 250 51 2.0 no
## 292 AB1487 c3k94 Small B 280 21 3.0 yes
## 293 AB1488 c3k95 Small B 209 30 1.0 yes
## 295 AB149 c3k97 Small B 118 70 0.5 no
## 303 AB1497 c4k6 Small B 330 27 4.0 yes
## 304 AB1498 c4k7 Small B 360 53 5.0 yes
## 312 AB1505 c4k15 Small B 345 12 5.0 yes
## 319 AB1511 c4k22 Small B 360 37 5.0 no
## 322 AB1514 c4k25 Small B 232 55 2.0 no
## 323 AB1515 c4k26 Small B 241 87 2.0 yes
## 326 AB1518 c4k29 Small B 375 34 6.0 yes
## 328 AB152 c4k31 Small B 193 81 1.0 no
## 332 AB1523 c4k35 Small B 316 31 4.0 yes
## 339 AB153 c4k42 Small B 166 36 1.0 yes
## 341 AB1531 c4k44 Small B 295 49 3.0 yes
## 343 AB1533 c4k46 Small B 344 24 5.0 yes
## 344 AB1534 c4k47 Small B 234 71 2.0 no
## 346 AB1536 c4k49 Small B 249 19 2.0 yes
## 348 AB1538 c4k51 Small B 280 13 3.0 yes
## 351 AB1540 c4k54 Small B 300 82 3.0 no
## 352 AB1541 c4k55 Small B 310 54 4.0 yes
## 354 AB1543 c4k57 Small B 318 71 4.0 yes
## 358 AB1547 c4k61 Small B 258 68 2.0 yes
## 360 AB1549 c4k63 Small B 218 54 2.0 no
## 362 AB1550 c4k65 Small B 270 60 3.0 yes
## 363 AB1551 c4k66 Small B 286 89 3.0 no
## 367 AB1555 c4k70 Small B 268 80 3.0 no
## 369 AB1557 c4k72 Small B 290 89 3.0 yes
## 375 AB1562 c4k78 Small B 296 45 3.0 yes
## 380 AB1567 c4k83 Small B 270 48 3.0 no
## 383 AB157 c4k86 Small B 185 52 1.0 yes
## 386 AB1572 c4k89 Small B 270 76 3.0 no
## 391 AB1577 c4k94 Small B 313 46 4.0 yes
## 394 AB158 c4k97 Small B 76 85 0.5 yes
## 395 AB1580 c4k98 Small B 299 85 3.0 no
## 401 AB1586 c5k5 Small B 261 40 2.0 no
## 405 AB159 c5k9 Small B 65 73 0.5 yes
## 406 AB1590 c5k10 Small B 304 26 4.0 yes
## 409 AB1593 c5k13 Small B 293 81 3.0 yes
## 414 AB1598 c5k18 Small B 292 89 3.0 no
## 420 AB1603 c5k24 Small B 226 30 2.0 yes
## 421 AB1604 c5k25 Small B 315 27 4.0 yes
## 423 AB1606 c5k27 Small B 279 57 3.0 yes
## 424 AB1607 c5k28 Small B 304 66 4.0 yes
## 429 AB1611 c5k33 Small B 213 38 1.0 no
## 430 AB1612 c5k34 Small B 230 14 2.0 no
## 431 AB1613 c5k35 Small B 319 83 4.0 no
## 435 AB1617 c5k39 Small B 302 22 4.0 yes
## 442 AB1623 c5k46 Small B 341 52 5.0 yes
## 445 AB1626 c5k49 Small B 331 16 4.0 no
## 450 AB1630 c5k54 Small B 236 31 2.0 yes
## 451 AB1631 c5k55 Small B 226 18 2.0 no
## 461 AB1640 c5k65 Small B 280 41 3.0 yes
## 464 AB1643 c5k68 Small B 233 55 2.0 no
## 467 AB1646 c5k71 Small B 334 76 5.0 no
## 470 AB1649 c5k74 Small B 234 11 2.0 yes
## 471 AB165 c5k75 Small B 149 19 0.5 yes
## 472 AB1650 c5k76 Small B 236 59 2.0 yes
## 473 AB1651 c5k77 Small B 182 28 1.0 yes
## 477 AB1655 c5k81 Small B 305 86 4.0 yes
## 479 AB1657 c5k83 Small B 251 12 2.0 yes
## 480 AB1658 c5k84 Small B 280 25 3.0 yes
## 481 AB1659 c5k85 Small B 278 53 3.0 yes
## 485 AB1662 c5k89 Small B 243 27 2.0 yes
## 486 AB1663 c5k90 Small B 292 57 3.0 no
## 490 AB1667 c5k94 Small B 299 89 3.0 no
## 491 AB1668 c5k95 Small B 336 80 5.0 yes
## 500 AB1676 c6k5 Small B 296 45 3.0 yes
## 502 AB1678 c6k7 Small B 235 27 2.0 yes
## 503 AB1679 c6k8 Small B 250 59 2.0 yes
## 504 AB168 c6k9 Small B 87 9 0.5 yes
## 505 AB1680 c6k10 Small B 235 91 2.0 yes
## 509 AB1684 c6k14 Small B 340 44 5.0 no
## 511 AB1686 c6k16 Small B 269 36 3.0 yes
## 512 AB1687 c6k17 Small B 315 63 4.0 yes
## 520 AB1694 c6k25 Small B 170 52 1.0 no
## 527 AB1700 c6k32 Small B 217 78 2.0 yes
## 533 AB1706 c6k38 Small B 123 34 0.5 yes
## 540 AB1712 c6k45 Small B 172 48 1.0 yes
## 553 AB1724 c6k58 Small B 195 45 1.0 no
## 554 AB1725 c6k59 Small B 130 74 0.5 no
## 555 AB1726 c6k60 Small B 80 69 0.5 no
## 560 AB1730 c6k65 Small B 200 65 1.0 no
## 576 AB1745 c6k81 Small B 110 30 0.5 no
## 580 AB1749 c6k85 Small B 179 48 1.0 yes
## 582 AB1750 c6k87 Small B 164 72 1.0 yes
## 587 AB1755 c6k92 Small B 135 47 0.5 no
## 591 AB1759 c6k96 Small B 195 85 1.0 no
## 595 AB1762 c7k1 Small B 188 85 1.0 no
## 597 AB1764 c7k3 Small B 189 53 1.0 no
## 599 AB1766 c7k5 Small B 155 75 1.0 yes
## 601 AB1768 c7k7 Small B 190 25 1.0 yes
## 607 AB1773 c7k13 Small B 188 73 1.0 no
## 615 AB1780 c7k21 Small B 158 23 1.0 no
## 623 AB1788 c7k29 Small B 126 58 0.5 yes
## 627 AB1791 c7k33 Small B 97 21 0.5 yes
## 631 AB1795 c7k37 Small B 131 46 0.5 no
## 634 AB1798 c7k40 Small B 97 45 0.5 yes
## 639 AB1802 c7k45 Small B 131 82 0.5 no
## 640 AB1803 c7k46 Small B 141 63 0.5 no
## 648 AB1810 c7k54 Small B 156 23 1.0 yes
## 656 AB1818 c7k62 Small B 151 7 1.0 yes
## 658 AB182 c7k64 Small B 87 5 0.5 yes
## 660 AB1821 c7k66 Small B 80 41 0.5 yes
## 663 AB1824 c7k69 Small B 209 74 1.0 no
## 664 AB1825 c7k70 Small B 181 64 1.0 yes
## 668 AB1829 c7k74 Small B 111 34 0.5 yes
## 670 AB1830 c7k76 Small B 117 38 0.5 no
## 673 AB1833 c7k79 Small B 152 83 1.0 yes
## 674 AB1834 c7k80 Small B 131 22 0.5 no
## 681 AB1840 c7k87 Small B 185 64 1.0 yes
## 686 AB1845 c7k92 Small B 134 7 0.5 no
## 689 AB1848 c7k95 Small B 179 24 1.0 yes
## 694 AB1852 c8k1 Small C 166 40 1.0 yes
## 695 AB1853 c8k2 Small C 190 21 1.0 yes
## 697 AB1855 c8k4 Small C 131 74 0.5 yes
## 700 AB1858 c8k7 Small C 176 20 1.0 no
## 702 AB186 c8k9 Small C 67 77 0.5 no
## 711 AB1868 c8k18 Small C 183 12 1.0 no
## 713 AB187 c8k20 Small C 98 65 0.5 yes
## 717 AB1873 c8k24 Small C 171 60 1.0 yes
## 720 AB1876 c8k27 Small C 135 51 0.5 yes
## 724 AB188 c8k31 Small C 117 82 0.5 yes
## 725 AB1880 c8k32 Small C 319 15 4.0 yes
## 726 AB1881 c8k33 Small C 450 54 9.0 no
## 734 AB1889 c8k41 Small C 394 79 7.0 yes
## 739 AB1893 c8k46 Small C 391 71 7.0 yes
## 742 AB1896 c8k49 Small C 319 67 4.0 yes
## 747 AB1900 c8k54 Small C 430 17 8.0 yes
## 751 AB1904 c8k58 Small C 450 50 9.0 yes
## 753 AB1907 c8k60 Small C 470 52 10.0 no
## 755 AB191 c8k62 Small C 76 37 0.5 yes
## 758 AB1912 c8k65 Small C 477 33 10.0 yes
## 759 AB1913 c8k66 Small C 368 77 6.0 yes
## 764 AB1918 c8k71 Small C 390 39 7.0 yes
## 766 AB192 c8k73 Small C 198 13 1.0 no
## 767 AB1920 c8k74 Small C 440 58 8.0 yes
## 768 AB1921 c8k75 Small C 420 88 7.0 no
## 770 AB1923 c8k77 Small C 450 58 9.0 yes
## 777 AB1930 c8k84 Small C 380 34 6.0 yes
## 779 AB1932 c8k86 Small C 420 92 7.0 yes
## 789 AB1944 c8k96 Small C 380 38 6.0 yes
## 795 AB195 c9k3 Small C 198 49 1.0 yes
## 797 AB1951 c9k5 Small C 480 33 10.0 yes
## 801 AB1955 c9k9 Small C 440 82 8.0 no
## 807 AB1961 c9k15 Small C 430 21 8.0 no
## 809 AB1963 c9k17 Small C 400 95 7.0 no
## 815 AB1969 c9k23 Small C 389 83 6.0 yes
## 817 AB1970 c9k25 Small C 356 65 5.0 no
## 819 AB1972 c9k27 Small C 400 31 7.0 yes
## 824 AB1977 c9k32 Small C 390 31 6.0 no
## 825 AB1978 c9k33 Small C 471 101 10.0 no
## 827 AB198 c9k35 Small C 119 38 0.5 yes
## 828 AB1980 c9k36 Small C 363 45 5.0 yes
## 830 AB1982 c9k38 Small C 440 74 8.0 yes
## 837 AB199 c9k45 Small C 137 35 0.5 yes
## 845 AB1997 c9k53 Small C 400 95 7.0 yes
## 848 AB200 c9k56 Small C 155 11 1.0 no
## 851 AB2002 c9k59 Small C 432 37 8.0 yes
## 852 AB2003 c9k60 Small C 480 93 10.0 no
## 857 AB2009 c9k65 Small C 491 74 10.5 yes
## 858 AB201 c9k66 Small C 150 7 1.0 yes
## 859 AB2010 c9k67 Small C 451 79 9.0 yes
## 861 AB2016 c9k69 Small C 436 97 8.0 yes
## 862 AB2017 c9k70 Small C 363 97 5.0 no
## 865 AB2021 c9k73 Small C 324 23 4.0 no
## 866 AB2022 c9k74 Small C 370 54 6.0 no
## 870 AB2026 c9k78 Small C 460 39 9.0 yes
## 873 AB203 c9k81 Small C 102 29 0.5 yes
## 877 AB2034 c9k85 Small C 434 53 8.0 yes
## 878 AB2035 c9k86 Small C 458 47 9.0 yes
## 879 AB2036 c9k87 Small C 490 54 10.5 yes
## 883 AB204 c9k91 Small C 91 65 0.5 yes
## 886 AB2043 c9k94 Small C 400 79 7.0 yes
## 887 AB2044 c9k95 Small C 404 87 7.0 yes
## 890 AB2049 c9k98 Small C 480 85 10.0 yes
## 898 AB2059 c10k7 Small C 417 20 7.0 no
## 899 AB206 c10k8 Small C 110 62 0.5 no
## 901 AB2062 c10k10 Small C 333 76 5.0 no
## 902 AB2063 c10k11 Small C 380 74 6.0 no
## 908 AB207 c10k17 Small C 171 12 1.0 yes
## 910 AB2071 c10k19 Small C 375 30 6.0 no
## 918 AB208 c10k27 Small C 120 70 0.5 yes
## 921 AB2082 c10k30 Small C 480 41 10.0 yes
## 926 AB209 c10k35 Small C 96 49 0.5 no
## 927 AB2090 c10k36 Small C 480 45 10.0 yes
## 931 AB2095 c10k40 Small C 361 33 5.0 yes
## 936 AB2100 c10k45 Small C 400 91 7.0 no
## 937 AB2101 c10k46 Small C 400 39 7.0 yes
## 939 AB2103 c10k48 Small C 308 82 4.0 no
## 941 AB2105 c10k50 Small C 416 92 7.0 no
## 943 AB2108 c10k52 Small C 410 48 7.0 no
## 946 AB2111 c10k55 Small C 480 77 10.0 no
## 951 AB2118 c10k60 Small C 480 85 10.0 no
## 952 AB2119 c10k61 Small C 480 37 10.0 yes
## 953 AB212 c10k62 Small C 185 44 1.0 no
## 954 AB2120 c10k63 Small C 350 20 5.0 yes
## 955 AB2121 c10k64 Small C 450 26 9.0 yes
## 963 AB2130 c10k72 Small C 460 67 9.0 no
## 966 AB2133 c10k75 Small C 370 66 6.0 no
## 967 AB2134 c10k76 Small C 360 29 5.0 yes
## 968 AB2135 c10k77 Small C 450 90 9.0 yes
## 974 AB2142 c10k83 Small C 390 19 7.0 yes
## 978 AB2146 c10k87 Small C 476 81 10.0 yes
## 981 AB2149 c10k90 Small C 410 60 7.0 no
## 987 AB2155 c10k96 Small C 480 41 10.0 no
## 989 AB2157 c10k98 Small C 485 61 10.5 yes
## 999 AB2166 c11k9 Small C 392 75 7.0 yes
## 1003 AB217 c11k13 Small C 64 69 0.5 yes
## 1012 AB2181 c11k22 Small C 347 60 5.0 yes
## 1013 AB2182 c11k23 Small C 370 46 6.0 yes
## 1016 AB2186 c11k26 Small C 345 40 5.0 yes
## 1019 AB2189 c11k29 Small C 410 60 7.0 no
## 1023 AB2194 c11k33 Small C 479 69 10.0 no
## 1024 AB2195 c11k34 Small C 445 38 9.0 yes
## 1032 AB2203 c11k42 Small C 421 85 8.0 yes
## 1042 AB2213 c11k52 Small C 400 47 7.0 yes
## 1043 AB2215 c11k53 Small C 466 32 10.0 yes
## 1046 AB2218 c11k56 Small C 365 85 6.0 no
## 1047 AB2219 c11k57 Small C 350 25 5.0 yes
## 1049 AB2220 c11k59 Small C 402 39 7.0 yes
## 1051 AB2223 c11k61 Small C 360 17 5.0 no
## 1052 AB2224 c11k62 Small C 361 33 5.0 yes
## 1053 AB2225 c11k63 Small C 350 36 5.0 yes
## 1057 AB2230 c11k67 Small C 343 80 5.0 no
## 1064 AB2237 c11k74 Small C 487 53 10.5 no
## 1069 AB2242 c11k79 Small C 418 44 7.0 no
## 1071 AB2244 c11k81 Small C 491 94 10.5 no
## 1074 AB2247 c11k84 Small C 320 35 4.0 yes
## 1083 AB2258 c11k93 Small C 394 87 7.0 yes
## 1084 AB2259 c11k94 Small C 463 24 10.0 yes
## 1086 AB2260 c11k96 Small C 443 30 8.0 no
## 1087 AB2261 c11k97 Small C 440 70 8.0 no
## 1098 AB2274 c12k9 Small C 443 74 8.0 no
## 1100 AB2276 c12k11 Small C 410 88 7.0 yes
## 1101 AB2277 c12k12 Small C 370 33 6.0 yes
## 1105 AB2280 c12k16 Small C 430 61 8.0 yes
## 1113 AB2289 c12k24 Small C 440 82 8.0 yes
## 1116 AB2291 c12k27 Small C 410 52 7.0 no
## 1118 AB2293 c12k29 Small C 450 79 9.0 yes
## 1119 AB2294 c12k30 Small C 460 27 9.0 no
## 1121 AB2296 c12k32 Small C 461 44 9.0 no
## 1125 AB230 c12k36 Small C 158 15 1.0 yes
## 1130 AB2304 c12k41 Small C 360 73 5.0 yes
## 1134 AB2310 c12k45 Small C 324 31 4.0 no
## 1135 AB2311 c12k46 Small C 393 67 7.0 yes
## 1136 AB2312 c12k47 Small C 424 61 8.0 yes
## 1140 AB2316 c12k51 Small C 392 47 7.0 yes
## 1141 AB2317 c12k52 Small C 433 97 8.0 yes
## 1143 AB2319 c12k54 Small C 419 100 7.0 yes
## 1144 AB232 c12k55 Small C 87 9 0.5 yes
## 1150 AB2325 c12k61 Small C 404 79 7.0 yes
## 1153 AB2329 c12k64 Small C 427 85 8.0 no
## 1154 AB233 c12k65 Small C 111 14 0.5 no
## 1158 AB2333 c12k69 Small C 364 57 6.0 yes
## 1159 AB2335 c12k70 Small C 380 50 6.0 no
## 1164 AB2341 c12k75 Small C 431 33 8.0 yes
## 1172 AB2349 c12k83 Small C 374 34 6.0 yes
## 1173 AB235 c12k84 Small C 65 37 0.5 yes
## 1186 AB2362 c12k97 Small C 382 66 6.0 yes
## 1187 AB2363 c12k98 Small C 449 58 9.0 yes
## 1192 AB2368 c13k4 Small C 438 61 8.0 yes
## 1193 AB2369 c13k5 Small C 460 51 9.0 yes
## 1199 AB2374 c13k11 Small C 359 41 5.0 yes
## 1201 AB2377 c13k13 Small C 365 81 6.0 yes
## 1207 AB2385 c13k19 Small C 375 46 6.0 yes
## 1209 AB2387 c13k21 Small C 405 52 7.0 no
## 1214 AB2391 c13k26 Small C 369 29 6.0 no
## 1215 AB2392 c13k27 Small C 329 31 4.0 no
## 1219 AB240 c13k31 Small C 98 29 0.5 yes
## 1220 AB241 c13k32 Small C 80 65 0.5 yes
## 1223 AB244 c13k35 Small C 121 30 0.5 no
## 1235 AB256 c13k47 Small C 85 25 0.5 no
## 1246 AB267 c13k58 Small C 121 42 0.5 yes
## 1247 AB268 c13k59 Small C 116 78 0.5 no
## 1251 AB272 c13k63 Small C 112 70 0.5 yes
## 1255 AB276 c13k67 Small C 119 38 0.5 yes
## 1256 AB277 c13k68 Small C 130 46 0.5 yes
## 1258 AB279 c13k70 Small C 152 23 1.0 no
## 1261 AB282 c13k73 Small C 52 13 0.5 no
## 1265 AB286 c13k77 Small C 143 83 0.5 yes
## 1268 AB289 c13k80 Small C 156 27 1.0 yes
## 1269 AB290 c13k81 Small C 144 47 0.5 no
## 1270 AB291 c13k82 Small C 168 16 1.0 yes
## 1281 AB302 c13k93 Small C 204 53 1.0 yes
## 1285 AB306 c13k97 Small C 148 79 0.5 no
## 1290 AB311 c14k3 Small C 122 42 0.5 yes
## 1309 AB330 c14k22 Small C 138 7 0.5 yes
## 1311 AB332 c14k24 Small C 172 24 1.0 yes
## 1313 AB334 c14k26 Small C 162 36 1.0 no
## 1316 AB337 c14k29 Small C 131 74 0.5 no
## 1325 AB346 c14k38 Small C 117 62 0.5 yes
## 1326 AB347 c14k39 Small C 136 27 0.5 yes
## 1329 AB350 c14k42 Small C 139 79 0.5 yes
## 1333 AB354 c14k46 Small C 174 32 1.0 yes
## 1336 AB357 c14k49 Small C 200 45 1.0 no
## 1341 AB362 c14k54 Small C 191 45 1.0 yes
## 1342 AB363 c14k55 Small C 177 56 1.0 no
## 1349 AB370 c14k62 Small C 185 8 1.0 yes
## 1350 AB371 c14k63 Small C 120 10 0.5 no
## 1352 AB373 c14k65 Small C 99 21 0.5 yes
## 1354 AB375 c14k67 Small C 111 6 0.5 yes
## 1355 AB376 c14k68 Small C 109 42 0.5 yes
## 1357 AB378 c14k70 Small C 97 5 0.5 no
## 1360 AB381 c14k73 Small C 117 66 0.5 no
## 1362 AB383 c14k75 Small C 144 11 0.5 no
## 1363 AB384 c14k76 Small C 188 44 1.0 yes
## 1364 AB385 c14k77 Small C 214 38 1.0 no
## 1367 AB388 c14k80 Small C 96 29 0.5 yes
## 1368 AB389 c14k81 Small C 120 6 0.5 no
## 1369 AB390 c14k82 Small C 114 26 0.5 no
## 1371 AB392 c14k84 Small C 124 62 0.5 yes
## 1373 AB394 c14k86 Small C 132 82 0.5 yes
## 1378 AB399 c14k91 Small C 169 68 1.0 no
## 1380 AB401 c14k93 Small C 188 73 1.0 yes
## 1382 AB403 c14k95 Small C 169 48 1.0 yes
## 1396 AB417 c15k10 Small C 149 27 0.5 yes
## 1403 AB424 c15k17 Small C 230 39 2.0 yes
## 1408 AB429 c15k22 Small C 270 60 3.0 no
## 1412 AB433 c15k26 Small C 232 87 2.0 no
## 1421 AB442 c15k35 Small C 321 27 4.0 yes
## 1422 AB443 c15k36 Small C 249 67 2.0 no
## 1426 AB447 c15k40 Small C 250 91 2.0 yes
## 1428 AB449 c15k42 Small C 197 41 1.0 no
## 1429 AB450 c15k43 Small C 292 29 3.0 yes
## 1430 AB451 c15k44 Small C 286 13 3.0 yes
## 1432 AB453 c15k46 Small C 262 56 2.0 yes
## 1438 AB459 c15k52 Small C 365 29 6.0 yes
## 1440 AB461 c15k54 Small C 292 13 3.0 yes
## 1441 AB462 c15k55 Small C 316 43 4.0 no
## 1443 AB464 c15k57 Small C 252 28 2.0 yes
## 1444 AB465 c15k58 Small C 233 67 2.0 yes
## 1447 AB468 c15k61 Small C 272 40 3.0 no
## 1451 AB472 c15k65 Small C 221 66 2.0 yes
## 1452 AB473 c15k66 Small C 247 91 2.0 no
## 1453 AB474 c15k67 Small C 251 51 2.0 yes
## 1459 AB480 c15k73 Small C 320 71 4.0 yes
## 1461 AB482 c15k75 Small C 251 11 2.0 yes
## 1464 AB485 c15k78 Small C 237 23 2.0 yes
## 1472 AB493 c15k86 Small C 290 41 3.0 no
## 1473 AB494 c15k87 Small C 292 17 3.0 no
## 1474 AB495 c15k88 Small C 324 75 4.0 yes
## 1477 AB498 c15k91 Small C 343 84 5.0 yes
## 1480 AB501 c15k94 Small C 275 44 3.0 no
## 1483 AB504 c15k97 Small C 305 38 4.0 no
## 1487 AB508 c16k2 Small C 383 26 6.0 no
## 1488 AB509 c16k3 Small C 226 38 2.0 yes
## 1491 AB512 c16k6 Small C 250 75 2.0 yes
## 1494 AB515 c16k9 Small C 274 44 3.0 no
## 1498 AB519 c16k13 Small C 280 21 3.0 yes
## 1501 AB522 c16k16 Small C 248 83 2.0 yes
## 1505 AB526 c16k20 Small C 263 24 3.0 yes
## 1509 AB530 c16k24 Small C 328 15 4.0 no
## 1515 AB536 c16k30 Small C 190 65 1.0 no
## 1516 AB537 c16k31 Small C 242 23 2.0 no
## 1517 AB538 c16k32 Small C 218 22 2.0 no
## 1523 AB544 c16k38 Small C 341 16 5.0 yes
## 1526 AB547 c16k41 Small C 202 45 1.0 yes
## 1528 AB549 c16k43 Small C 234 11 2.0 yes
## 1530 AB551 c16k45 Small C 212 10 1.0 no
## 1532 AB553 c16k47 Small C 308 14 4.0 no
## 1534 AB555 c16k49 Small C 279 85 3.0 yes
## 1540 AB561 c16k55 Small C 191 45 1.0 yes
## 1544 AB565 c16k59 Small C 262 16 2.0 yes
## 1549 AB591 c16k64 Small C 460 83 9.0 no
## 1550 AB599 c16k65 Small C 490 26 10.5 no
## 1551 AB604 c16k66 Small C 494 26 10.5 no
## 1554 AB639 c16k69 Small D 470 24 10.0 no
## 1557 AB649 c16k72 Small D 470 24 10.0 no
## 1558 AB661 c16k73 Small D 460 96 9.0 no
## 1561 AB679 c16k76 Small D 490 50 10.5 yes
## 1562 AB703 c16k77 Small D 470 68 10.0 yes
## 1570 AB811 c16k85 Small D 480 73 10.0 yes
## 1571 AB827 c16k86 Small D 470 84 10.0 yes
## 1587 AB096 c17k3 Medium A 741 90 27.0 yes
## 1591 AB1003 c17k7 Medium A 759 112 29.0 no
## 1602 AB1015 c17k18 Medium A 834 140 35.0 yes
## 1604 AB1017 c17k20 Medium A 553 92 14.0 yes
## 1605 AB1018 c17k21 Medium A 570 57 15.0 no
## 1606 AB1019 c17k22 Medium A 694 53 24.0 no
## 1607 AB102 c17k23 Medium A 550 67 14.0 yes
## 1608 AB1020 c17k24 Medium A 730 126 27.0 no
## 1615 AB103 c17k31 Medium A 662 50 22.0 yes
## 1620 AB1034 c17k36 Medium A 790 54 31.0 yes
## 1628 AB1047 c17k44 Medium A 601 32 18.0 no
## 1630 AB1049 c17k46 Medium A 672 39 23.0 yes
## 1634 AB1052 c17k50 Medium A 730 117 27.0 no
## 1638 AB1056 c17k54 Medium A 647 93 21.0 yes
## 1642 AB1060 c17k58 Medium A 705 107 25.0 yes
## 1643 AB1061 c17k59 Medium A 574 73 15.0 no
## 1644 AB1062 c17k60 Medium A 664 110 22.0 no
## 1647 AB1065 c17k63 Medium A 704 55 25.0 no
## 1648 AB1066 c17k64 Medium A 850 102 36.0 yes
## 1650 AB1068 c17k66 Medium A 872 87 38.0 no
## 1651 AB1069 c17k67 Medium A 937 127 43.0 yes
## 1652 AB107 c17k68 Medium A 560 72 15.0 yes
## 1654 AB1071 c17k70 Medium A 980 75 46.0 yes
## 1655 AB1073 c17k71 Medium A 899 93 40.0 yes
## 1656 AB1074 c17k72 Medium A 534 37 13.0 yes
## 1657 AB1075 c17k73 Medium A 563 64 15.0 no
## 1658 AB1076 c17k74 Medium A 590 47 17.0 yes
## 1662 AB108 c17k78 Medium A 635 40 20.0 no
## 1667 AB1085 c17k83 Medium A 585 59 16.0 no
## 1668 AB1086 c17k84 Medium A 589 91 16.0 yes
## 1671 AB1089 c17k87 Medium A 740 126 27.0 yes
## 1674 AB1091 c17k90 Medium A 704 103 25.0 no
## 1678 AB1099 c17k94 Medium A 580 97 16.0 yes
## 1680 AB1100 c17k96 Medium A 560 52 15.0 no
## 1681 AB1101 c17k97 Medium A 610 53 18.0 no
## 1682 AB1102 c17k98 Medium A 580 62 16.0 no
## 1685 AB1105 c18k2 Medium A 680 80 23.0 yes
## 1686 AB1106 c18k3 Medium A 710 91 25.0 yes
## 1687 AB1107 c18k4 Medium A 837 124 35.0 no
## 1697 AB112 c18k14 Medium A 780 81 30.0 no
## 1701 AB1127 c18k18 Medium A 712 56 26.0 no
## 1702 AB1128 c18k19 Medium A 660 54 22.0 no
## 1704 AB1131 c18k21 Medium A 627 71 19.0 yes
## 1705 AB1133 c18k22 Medium A 570 65 15.0 yes
## 1707 AB1136 c18k24 Medium A 700 51 25.0 yes
## 1709 AB1138 c18k26 Medium A 637 104 20.0 no
## 1720 AB1149 c18k37 Medium A 540 38 13.0 no
## 1723 AB1152 c18k40 Medium A 560 60 15.0 no
## 1724 AB1154 c18k41 Medium A 525 57 13.0 no
## 1730 AB1160 c18k47 Medium A 850 113 36.0 no
## 1734 AB1164 c18k51 Medium A 753 56 29.0 yes
## 1735 AB1165 c18k52 Medium A 760 68 29.0 no
## 1738 AB1168 c18k55 Medium A 610 69 18.0 no
## 1747 AB1178 c18k64 Medium A 668 63 22.0 yes
## 1757 AB1189 c18k74 Medium B 542 86 14.0 no
## 1764 AB1197 c18k81 Medium B 730 81 27.0 no
## 1771 AB1204 c18k88 Medium B 820 58 33.0 yes
## 1773 AB1207 c18k90 Medium B 920 118 41.0 yes
## 1779 AB1216 c18k96 Medium B 632 112 20.0 no
## 1781 AB1221 c18k98 Medium B 500 75 11.0 yes
## 1783 AB1223 c19k1 Medium B 637 108 20.0 no
## 1784 AB1225 c19k2 Medium B 560 44 15.0 yes
## 1787 AB1230 c19k5 Medium B 600 56 17.0 no
## 1789 AB1234 c19k7 Medium B 704 47 25.0 yes
## 1790 AB1235 c19k8 Medium B 530 97 13.0 no
## 1791 AB1236 c19k9 Medium B 530 37 13.0 no
## 1797 AB1244 c19k15 Medium B 590 47 17.0 yes
## 1798 AB1245 c19k16 Medium B 720 117 27.0 no
## 1807 AB1256 c19k25 Medium B 618 70 19.0 yes
## 1816 AB1265 c19k34 Medium B 589 107 16.0 yes
## 1817 AB1266 c19k35 Medium B 671 111 23.0 no
## 1818 AB1267 c19k36 Medium B 940 119 43.0 yes
## 1827 AB1276 c19k45 Medium B 551 87 14.0 yes
## 1830 AB1279 c19k48 Medium B 567 72 15.0 yes
## 1831 AB1280 c19k49 Medium B 963 106 45.0 yes
## 1835 AB1285 c19k53 Medium B 669 79 22.0 yes
## 1838 AB1288 c19k56 Medium B 650 85 21.0 yes
## 1843 AB1294 c19k61 Medium B 605 81 18.0 yes
## 1845 AB1296 c19k63 Medium B 674 47 23.0 yes
## 1846 AB1297 c19k64 Medium B 570 37 15.0 yes
## 1849 AB1300 c19k67 Medium B 896 117 40.0 yes
## 1850 AB1301 c19k68 Medium B 620 67 19.0 yes
## 1851 AB1302 c19k69 Medium B 944 131 43.0 no
## 1852 AB1303 c19k70 Medium B 503 91 12.0 yes
## 1855 AB1307 c19k73 Medium B 755 124 29.0 no
## 1858 AB1311 c19k76 Medium B 623 87 19.0 yes
## 1860 AB1313 c19k78 Medium B 697 65 24.0 yes
## 1864 AB1318 c19k82 Medium B 511 104 12.0 yes
## 1865 AB1319 c19k83 Medium B 540 74 13.0 no
## 1867 AB1321 c19k85 Medium B 619 94 19.0 yes
## 1876 AB1330 c19k94 Medium B 672 59 23.0 yes
## 1878 AB1905 c19k96 Medium C 500 59 12.0 no
## 1883 AB1942 c20k2 Medium C 560 84 14.0 yes
## 1885 AB1987 c20k4 Medium C 525 65 13.0 yes
## 1889 AB2014 c20k8 Medium C 530 25 13.0 yes
## 1890 AB2015 c20k9 Medium C 520 64 12.0 no
## 1891 AB2019 c20k10 Medium C 530 89 13.0 yes
## 1895 AB2042 c20k14 Medium C 510 80 12.0 yes
## 1901 AB2061 c20k20 Medium C 510 84 12.0 no
## 1913 AB2125 c20k32 Medium C 505 79 12.0 no
## 1914 AB2127 c20k33 Medium C 500 75 12.0 no
## 1917 AB2154 c20k36 Medium C 504 87 12.0 yes
## 1920 AB2179 c20k39 Medium C 579 49 16.0 yes
## 1922 AB2185 c20k41 Medium C 500 83 11.0 yes
## 1924 AB2192 c20k43 Medium C 530 41 13.0 no
## 1930 AB2228 c20k49 Medium C 502 103 12.0 yes
## 1932 AB2251 c20k51 Medium C 505 51 12.0 yes
## 1937 AB2272 c20k56 Medium C 571 65 15.0 yes
## 1940 AB2307 c20k59 Medium C 578 33 16.0 yes
## 1941 AB2308 c20k60 Medium C 577 101 16.0 yes
## 1943 AB2334 c20k62 Medium C 550 47 14.0 yes
## 1949 AB2379 c20k68 Medium C 591 39 17.0 yes
## 1952 AB570 c20k71 Medium C 530 49 13.0 no
## 1960 AB578 c20k79 Medium C 600 112 18.0 yes
## 1967 AB585 c20k86 Medium C 620 79 19.0 no
## 1968 AB586 c20k87 Medium C 620 99 19.0 no
## 1971 AB589 c20k90 Medium C 580 70 16.0 yes
## 1972 AB592 c20k91 Medium C 610 97 18.0 no
## 1976 AB596 c20k95 Medium C 570 77 15.0 no
## 1979 AB600 c20k98 Medium C 620 47 19.0 no
## 1981 AB602 c22k1 Medium C 840 89 35.0 yes
## 1983 AB605 c22k3 Medium C 619 62 19.0 yes
## 1984 AB606 c22k4 Medium C 500 39 12.0 yes
## 1989 AB612 c22k9 Medium D 640 61 21.0 no
## 1992 AB615 c22k12 Medium D 920 142 41.0 yes
## 1998 AB621 c22k18 Medium D 590 55 16.0 yes
## 2002 AB626 c22k22 Medium D 580 109 16.0 no
## 2004 AB628 c22k24 Medium D 650 93 21.0 yes
## 2005 AB629 c22k25 Medium D 650 61 21.0 yes
## 2011 AB635 c22k31 Medium D 828 95 34.0 yes
## 2017 AB646 c22k37 Medium D 610 69 18.0 yes
## 2019 AB648 c22k39 Medium D 580 74 16.0 no
## 2020 AB650 c22k40 Medium D 650 101 21.0 no
## 2026 AB656 c22k46 Medium D 640 113 21.0 no
## 2031 AB662 c22k51 Medium D 590 107 17.0 no
## 2033 AB664 c22k53 Medium D 640 97 21.0 no
## 2034 AB665 c22k54 Medium D 690 108 23.0 yes
## 2035 AB666 c22k55 Medium D 720 92 26.0 yes
## 2038 AB671 c22k58 Medium D 511 68 12.0 yes
## 2039 AB672 c22k59 Medium D 500 99 11.0 no
## 2046 AB681 c22k66 Medium D 520 93 12.0 no
## 2047 AB682 c22k67 Medium D 650 81 22.0 yes
## 2049 AB684 c22k69 Medium D 630 51 20.0 yes
## 2050 AB685 c22k70 Medium D 676 72 23.0 no
## 2052 AB687 c22k72 Medium D 650 77 21.0 yes
## 2061 AB696 c22k81 Medium D 580 90 16.0 no
## 2064 AB699 c22k84 Medium D 690 117 24.0 yes
## 2066 AB701 c22k86 Medium D 630 104 20.0 yes
## 2067 AB702 c22k87 Medium D 520 85 12.0 no
## 2068 AB704 c22k88 Medium D 570 65 15.0 no
## 2077 AB713 c22k97 Medium D 590 55 16.0 yes
## 2083 AB719 c23k4 Medium D 630 103 19.0 yes
## 2084 AB720 c23k5 Medium D 730 129 27.0 yes
## 2085 AB721 c23k6 Medium D 740 94 27.0 yes
## 2087 AB723 c23k8 Medium D 900 97 40.0 yes
## 2092 AB732 c23k13 Medium D 570 73 15.0 no
## 2095 AB735 c23k16 Medium D 570 45 15.0 no
## 2101 AB743 c23k22 Medium D 940 103 43.0 no
## 2102 AB750 c23k23 Medium D 521 85 13.0 yes
## 2106 AB756 c23k27 Medium D 620 111 19.0 yes
## 2109 AB759 c23k30 Medium D 770 121 29.0 yes
## 2110 AB760 c23k31 Medium D 810 130 33.0 no
## 2111 AB761 c23k32 Medium D 810 96 32.0 yes
## 2115 AB765 c23k36 Medium D 920 130 42.0 yes
## 2116 AB769 c23k37 Medium D 590 87 17.0 no
## 2117 AB770 c23k38 Medium D 700 126 24.0 yes
## 2123 AB776 c23k44 Medium D 569 60 15.0 yes
## 2130 AB783 c23k51 Medium D 750 108 28.0 yes
## 2142 AB795 c23k63 Medium D 628 91 19.0 yes
## 2143 AB796 c23k64 Medium D 810 57 32.0 yes
## 2145 AB798 c23k66 Medium D 880 120 38.0 no
## 2154 AB810 c23k75 Medium D 520 105 12.0 yes
## 2158 AB815 c23k79 Medium D 582 98 16.0 yes
## 2159 AB816 c23k80 Medium D 620 54 19.0 yes
## 2169 AB826 c23k90 Medium D 750 75 28.0 no
## 2173 AB831 c23k94 Medium D 505 55 12.0 no
## 2177 AB836 c23k98 Medium E 590 79 17.0 no
## 2180 AB839 c24k2 Medium E 722 89 27.0 no
## 2182 AB841 c24k4 Medium E 665 42 22.0 no
## 2184 AB843 c24k6 Medium E 710 91 25.0 no
## 2190 AB849 c24k12 Medium E 866 78 37.0 yes
## 2191 AB850 c24k13 Medium E 564 32 15.0 yes
## 2194 AB854 c24k16 Medium E 750 51 28.0 yes
## 2195 AB855 c24k17 Medium E 830 64 35.0 yes
## 2197 AB857 c24k19 Medium E 850 141 36.0 yes
## 2206 AB869 c24k28 Medium E 650 114 22.0 yes
## 2207 AB870 c24k29 Medium E 712 48 26.0 yes
## 2217 AB882 c24k39 Medium E 510 99 12.0 no
## 2219 AB884 c24k41 Medium E 540 82 13.0 yes
## 2221 AB886 c24k43 Medium E 605 69 18.0 yes
## 2228 AB896 c24k50 Medium E 620 98 19.0 yes
## 2239 AB911 c24k61 Medium E 595 71 17.0 yes
## 2244 AB918 c24k66 Medium E 783 66 31.0 yes
## 2245 AB919 c24k67 Medium E 951 73 43.0 yes
## 2246 AB920 c24k68 Medium E 555 56 14.0 yes
## 2250 AB924 c24k72 Medium E 570 57 15.0 no
## 2251 AB925 c24k73 Medium E 610 41 18.0 no
## 2252 AB926 c24k74 Medium E 640 100 20.0 yes
## 2258 AB933 c24k80 Medium E 550 87 14.0 yes
## 2262 AB937 c24k84 Medium E 519 92 12.0 yes
## 2263 AB938 c24k85 Medium E 520 52 12.0 no
## 2267 AB942 c24k89 Medium E 610 86 18.0 no
## 2279 AB955 c25k2 Medium E 944 64 43.0 no
## 2281 AB960 c25k4 Medium E 591 51 17.0 yes
## 2283 AB963 c25k6 Medium E 590 79 16.0 yes
## 2284 AB964 c25k7 Medium E 540 66 13.0 yes
## 2286 AB966 c25k9 Medium E 548 46 14.0 yes
## 2287 AB967 c25k10 Medium E 652 98 22.0 yes
## 2292 AB972 c25k15 Medium E 790 99 31.0 no
## 2293 AB973 c25k16 Medium E 620 86 19.0 yes
## 2300 AB980 c25k23 Medium E 780 90 30.0 no
## 2306 AB992 c25k29 Medium E 610 93 18.0 yes
## 2309 AB995 c25k32 Medium E 550 51 14.0 no
## 2310 AB996 c25k33 Medium E 581 102 16.0 yes
## 2311 AB997 c25k34 Medium E 590 75 16.0 no
## 2312 AB998 c25k35 Medium E 655 82 22.0 no
## 2319 AB1029 c25k42 Big A 1110 97 58.0 yes
## 2321 AB1039 c25k44 Big A 1016 96 50.0 no
## 2324 AB1042 c25k47 Big A 1100 161 57.0 yes
## 2327 AB1093 c25k50 Big A 1063 146 54.0 yes
## 2328 AB1094 c25k51 Big A 1153 144 62.0 yes
## 2336 AB1126 c25k59 Big A 1614 159 138.0 yes
## 2338 AB1132 c25k61 Big A 2510 258 305.0 yes
## 2341 AB1172 c25k64 Big A 1440 133 84.0 yes
## 2345 AB1206 c25k68 Big B 1003 115 49.0 yes
## 2347 AB1211 c25k70 Big B 1050 142 52.0 yes
## 2354 AB1283 c25k77 Big B 1510 163 107.0 yes
## 2357 AB590 c25k80 Big C 1360 134 76.0 yes
## 2362 AB727 c25k85 Big D 1450 162 94.0 yes
## 2363 AB741 c25k86 Big D 1093 119 56.0 no
## 2365 AB744 c25k88 Big D 1370 182 77.0 no
## 2371 AB766 c25k94 Big D 1110 93 58.0 yes
## 2376 AB864 c25k99 Big E 1360 126 73.0 yes
## 2387 AB932 c26k11 Big E 1360 104 72.0 yes
## 2394 AB986 c26k18 Big E 1297 161 67.0 no
## 2396 AB988 c26k20 Big E 1860 253 176.0 yesa.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.SI(N,n,a.estimar)## N muestra.Income muestra.Employees muestra.Taxes
## Estimation 2396 1.028317e+06 1.488923e+05 28079.951220
## Standard Error 0 2.018039e+04 2.508035e+03 1458.738245
## CVE 0 1.962467e+00 1.684462e+00 5.194946
## DEFF NaN 1.000000e+00 1.000000e+00 1.0000009.2.4 Estimación en dominios
\(U_d{\subset}U\) tal que \(U{=\bigcup}_{d=1}^{D}{U_d}\)
Si \(k{\in}U_l\), entonces \(k{\not\in}U_d\) para \(d{\neq}l\)
Tamaño absoluto; \(\#(U_d)=N_d\)
Tamaño relativo; \(P_d=\frac{\#(U_d)}{\#(U)}=\frac{N_d}{N}\)
\[t_{y_{d}}={\sum}_{U_d}y_k\]
\[ {z}_{dk}(U_d)= \begin{cases} 1&\text{ si }k{\in}U_d\\ 0&\text{ si }k{\notin}U_d \end{cases} \]
Zd <- Domains(muestra$SPAM);Zd## no yes
## [1,] 0 1
## [2,] 0 1
## [3,] 1 0
## [4,] 1 0
## [5,] 0 1
## [6,] 0 1
## [7,] 0 1
## [8,] 0 1
## [9,] 0 1
## [10,] 0 1
## [11,] 0 1
## [12,] 0 1
## [13,] 0 1
## [14,] 1 0
## [15,] 0 1
## [16,] 0 1
## [17,] 1 0
## [18,] 1 0
## [19,] 1 0
## [20,] 1 0
## [21,] 0 1
## [22,] 0 1
## [23,] 0 1
## [24,] 1 0
## [25,] 0 1
## [26,] 1 0
## [27,] 1 0
## [28,] 1 0
## [29,] 0 1
## [30,] 1 0
## [31,] 0 1
## [32,] 0 1
## [33,] 0 1
## [34,] 0 1
## [35,] 1 0
## [36,] 0 1
## [37,] 1 0
## [38,] 0 1
## [39,] 0 1
## [40,] 1 0
## [41,] 0 1
## [42,] 0 1
## [43,] 0 1
## [44,] 0 1
## [45,] 0 1
## [46,] 1 0
## [47,] 1 0
## [48,] 0 1
## [49,] 0 1
## [50,] 0 1
## [51,] 0 1
## [52,] 0 1
## [53,] 0 1
## [54,] 0 1
## [55,] 0 1
## [56,] 0 1
## [57,] 0 1
## [58,] 0 1
## [59,] 1 0
## [60,] 0 1
## [61,] 0 1
## [62,] 0 1
## [63,] 0 1
## [64,] 0 1
## [65,] 0 1
## [66,] 0 1
## [67,] 0 1
## [68,] 1 0
## [69,] 0 1
## [70,] 1 0
## [71,] 0 1
## [72,] 0 1
## [73,] 0 1
## [74,] 1 0
## [75,] 1 0
## [76,] 0 1
## [77,] 0 1
## [78,] 0 1
## [79,] 0 1
## [80,] 1 0
## [81,] 1 0
## [82,] 0 1
## [83,] 0 1
## [84,] 0 1
## [85,] 0 1
## [86,] 0 1
## [87,] 1 0
## [88,] 1 0
## [89,] 0 1
## [90,] 0 1
## [91,] 1 0
## [92,] 0 1
## [93,] 0 1
## [94,] 0 1
## [95,] 1 0
## [96,] 1 0
## [97,] 0 1
## [98,] 0 1
## [99,] 1 0
## [100,] 0 1
## [101,] 0 1
## [102,] 0 1
## [103,] 0 1
## [104,] 1 0
## [105,] 0 1
## [106,] 0 1
## [107,] 1 0
## [108,] 0 1
## [109,] 0 1
## [110,] 0 1
## [111,] 1 0
## [112,] 0 1
## [113,] 1 0
## [114,] 1 0
## [115,] 0 1
## [116,] 0 1
## [117,] 1 0
## [118,] 0 1
## [119,] 1 0
## [120,] 0 1
## [121,] 0 1
## [122,] 1 0
## [123,] 1 0
## [124,] 0 1
## [125,] 0 1
## [126,] 0 1
## [127,] 1 0
## [128,] 0 1
## [129,] 0 1
## [130,] 0 1
## [131,] 0 1
## [132,] 1 0
## [133,] 1 0
## [134,] 1 0
## [135,] 0 1
## [136,] 0 1
## [137,] 1 0
## [138,] 0 1
## [139,] 1 0
## [140,] 0 1
## [141,] 1 0
## [142,] 1 0
## [143,] 0 1
## [144,] 0 1
## [145,] 0 1
## [146,] 0 1
## [147,] 0 1
## [148,] 0 1
## [149,] 0 1
## [150,] 0 1
## [151,] 0 1
## [152,] 1 0
## [153,] 1 0
## [154,] 0 1
## [155,] 0 1
## [156,] 0 1
## [157,] 0 1
## [158,] 0 1
## [159,] 0 1
## [160,] 1 0
## [161,] 0 1
## [162,] 0 1
## [163,] 1 0
## [164,] 0 1
## [165,] 0 1
## [166,] 0 1
## [167,] 1 0
## [168,] 1 0
## [169,] 1 0
## [170,] 1 0
## [171,] 1 0
## [172,] 0 1
## [173,] 0 1
## [174,] 1 0
## [175,] 1 0
## [176,] 1 0
## [177,] 1 0
## [178,] 0 1
## [179,] 0 1
## [180,] 1 0
## [181,] 1 0
## [182,] 0 1
## [183,] 0 1
## [184,] 1 0
## [185,] 0 1
## [186,] 1 0
## [187,] 1 0
## [188,] 0 1
## [189,] 0 1
## [190,] 0 1
## [191,] 0 1
## [192,] 1 0
## [193,] 0 1
## [194,] 0 1
## [195,] 1 0
## [196,] 0 1
## [197,] 1 0
## [198,] 0 1
## [199,] 1 0
## [200,] 0 1
## [201,] 0 1
## [202,] 0 1
## [203,] 0 1
## [204,] 1 0
## [205,] 1 0
## [206,] 1 0
## [207,] 0 1
## [208,] 0 1
## [209,] 0 1
## [210,] 0 1
## [211,] 0 1
## [212,] 1 0
## [213,] 0 1
## [214,] 0 1
## [215,] 0 1
## [216,] 0 1
## [217,] 0 1
## [218,] 1 0
## [219,] 0 1
## [220,] 0 1
## [221,] 0 1
## [222,] 0 1
## [223,] 1 0
## [224,] 0 1
## [225,] 1 0
## [226,] 0 1
## [227,] 0 1
## [228,] 0 1
## [229,] 0 1
## [230,] 0 1
## [231,] 0 1
## [232,] 1 0
## [233,] 1 0
## [234,] 1 0
## [235,] 0 1
## [236,] 1 0
## [237,] 0 1
## [238,] 1 0
## [239,] 1 0
## [240,] 0 1
## [241,] 0 1
## [242,] 0 1
## [243,] 0 1
## [244,] 0 1
## [245,] 1 0
## [246,] 0 1
## [247,] 1 0
## [248,] 0 1
## [249,] 0 1
## [250,] 0 1
## [251,] 0 1
## [252,] 1 0
## [253,] 1 0
## [254,] 1 0
## [255,] 0 1
## [256,] 0 1
## [257,] 0 1
## [258,] 0 1
## [259,] 0 1
## [260,] 0 1
## [261,] 0 1
## [262,] 0 1
## [263,] 0 1
## [264,] 1 0
## [265,] 1 0
## [266,] 1 0
## [267,] 1 0
## [268,] 0 1
## [269,] 1 0
## [270,] 0 1
## [271,] 0 1
## [272,] 1 0
## [273,] 0 1
## [274,] 0 1
## [275,] 1 0
## [276,] 0 1
## [277,] 1 0
## [278,] 1 0
## [279,] 1 0
## [280,] 1 0
## [281,] 1 0
## [282,] 0 1
## [283,] 1 0
## [284,] 0 1
## [285,] 0 1
## [286,] 1 0
## [287,] 1 0
## [288,] 0 1
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## [290,] 0 1
## [291,] 0 1
## [292,] 1 0
## [293,] 1 0
## [294,] 0 1
## [295,] 0 1
## [296,] 0 1
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## [300,] 1 0
## [301,] 1 0
## [302,] 0 1
## [303,] 0 1
## [304,] 0 1
## [305,] 0 1
## [306,] 1 0
## [307,] 0 1
## [308,] 0 1
## [309,] 1 0
## [310,] 0 1
## [311,] 0 1
## [312,] 1 0
## [313,] 1 0
## [314,] 1 0
## [315,] 1 0
## [316,] 0 1
## [317,] 0 1
## [318,] 0 1
## [319,] 1 0
## [320,] 1 0
## [321,] 1 0
## [322,] 0 1
## [323,] 0 1
## [324,] 0 1
## [325,] 0 1
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## [327,] 0 1
## [328,] 1 0
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## [330,] 0 1
## [331,] 0 1
## [332,] 1 0
## [333,] 0 1
## [334,] 0 1
## [335,] 0 1
## [336,] 0 1
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## [339,] 0 1
## [340,] 1 0
## [341,] 1 0
## [342,] 0 1
## [343,] 1 0
## [344,] 0 1
## [345,] 0 1
## [346,] 0 1
## [347,] 0 1
## [348,] 0 1
## [349,] 0 1
## [350,] 0 1
## [351,] 0 1
## [352,] 0 1
## [353,] 0 1
## [354,] 1 0
## [355,] 1 0
## [356,] 1 0
## [357,] 0 1
## [358,] 0 1
## [359,] 1 0
## [360,] 1 0
## [361,] 0 1
## [362,] 1 0
## [363,] 0 1
## [364,] 0 1
## [365,] 0 1
## [366,] 1 0
## [367,] 1 0
## [368,] 0 1
## [369,] 0 1
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## [371,] 0 1
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## [373,] 1 0
## [374,] 0 1
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## [376,] 0 1
## [377,] 1 0
## [378,] 1 0
## [379,] 0 1
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## [381,] 0 1
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## [383,] 1 0
## [384,] 0 1
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## [386,] 0 1
## [387,] 1 0
## [388,] 0 1
## [389,] 0 1
## [390,] 0 1
## [391,] 1 0
## [392,] 1 0
## [393,] 1 0
## [394,] 0 1
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## [396,] 0 1
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## [398,] 1 0
## [399,] 0 1
## [400,] 0 1
## [401,] 1 0
## [402,] 0 1
## [403,] 0 1
## [404,] 0 1
## [405,] 0 1
## [406,] 1 0
## [407,] 1 0
## [408,] 0 1
## [409,] 1 0
## [410,] 0 1
## [411,] 1 0
## [412,] 0 1
## [413,] 0 1
## [414,] 0 1
## [415,] 0 1
## [416,] 0 1
## [417,] 1 0
## [418,] 0 1
## [419,] 0 1
## [420,] 1 0
## [421,] 0 1
## [422,] 1 0
## [423,] 0 1
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## [426,] 0 1
## [427,] 1 0
## [428,] 1 0
## [429,] 0 1
## [430,] 0 1
## [431,] 1 0
## [432,] 1 0
## [433,] 1 0
## [434,] 0 1
## [435,] 0 1
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## [437,] 0 1
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## [440,] 1 0
## [441,] 1 0
## [442,] 1 0
## [443,] 1 0
## [444,] 0 1
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## [446,] 0 1
## [447,] 1 0
## [448,] 1 0
## [449,] 0 1
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## [455,] 1 0
## [456,] 1 0
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## [458,] 0 1
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## [554,] 0 1
## [555,] 0 1
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## [697,] 0 1\[ \begin{align} t_{y_{d}}&={\sum}_{U}z_{dk}y_{k}\\ &={\sum}_{U}y_{dk} \end{align} \]
\[N_{d}={\sum}_{U}z_{dk}\]
\[ \begin{align} \bar{y}_{U_d}&=\frac{{\sum}_{U}y_{dk}}{{\sum}_{U}z_{dk}}\\ &=\frac{t_{y_{d}}}{N_{d}} \end{align} \]
\[ \begin{align} \widehat{t}_{y_{d},\pi}&=\frac{N}{n}{\sum}_{s}y_{dk}\\ &=\frac{N}{n}{\sum}_{s_d}y_{k} \end{align} \]
\[ \begin{align} V_{MAS}\left(\widehat{t}_{y_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right){S}_{y_{d}U}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{MAS}\left(\widehat{t}_{y_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right){S}_{y_{d}s}^{2} \end{align} \]
a.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.SI(N,n,Zd[,"yes"]*a.estimar)## N muestra.Income muestra.Employees muestra.Taxes
## Estimation 2396 6.522345e+05 92749.606887 18734.863702
## Standard Error 0 2.355643e+04 3085.333343 1430.791569
## CVE 0 3.611651e+00 3.326519 7.637054
## DEFF NaN 1.000000e+00 1.000000 1.000000a.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.SI(N,n,Zd[,"no"]*a.estimar)## N muestra.Income muestra.Employees muestra.Taxes
## Estimation 2396 3.760826e+05 56142.714491 9345.087518
## Standard Error 0 1.876602e+04 2716.870011 661.438201
## CVE 0 4.989865e+00 4.839221 7.077924
## DEFF NaN 1.000000e+00 1.000000 1.000000\[ \begin{align} \widehat{N}_{y_{d},\pi}&=\frac{N}{n}{\sum}_{s}z_{dk}\\ &=\frac{N}{n}{\sum}_{s_d}z_{k} \end{align} \]
\[ \begin{align} V_{MAS}\left(\widehat{N}_{y_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right){S}_{z_{d}U}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{MAS}\left(\widehat{N}_{y_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right){S}_{z_{d}s}^{2} \end{align} \]
a.estimar <- data.frame(muestra$Income,muestra$Employees,muestra$Taxes);E.SI(N,n,Zd[,"yes"])## N y
## Estimation 2396 1498.789096
## Standard Error 0 37.014015
## CVE 0 2.469595
## DEFF NaN 1.000000a.estimar <- data.frame(muestra$Income,muestra$Employees,muestra$Taxes);E.SI(N,n,Zd[,"no"])## N y
## Estimation 2396 897.210904
## Standard Error 0 37.014015
## CVE 0 4.125453
## DEFF NaN 1.000000\[ \begin{align} \widehat{P}_{y_{d},\pi}&=\frac{1}{N}\frac{N}{n}{\sum}_{s}z_{dk}\\ &=\frac{1}{n}{\sum}_{s_d}z_{k} \end{align} \]
\[ \begin{align} V_{MAS}\left(\widehat{P}_{y_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\frac{1}{N^2}{S}_{z_{d}U}^{2}\\ &=\frac{1}{n}\left(1-\frac{n}{N}\right){S}_{z_{d}U}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{MAS}\left(\widehat{P}_{y_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\frac{1}{N^2}{S}_{z_{d}s}^{2}\\ &=\frac{1}{n}\left(1-\frac{n}{N}\right){S}_{z_{d}s}^{2} \end{align} \]
a.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.SI(N,n,(1/N)*Zd[,"yes"])## N y
## Estimation 2396 0.62553802
## Standard Error 0 0.01544825
## CVE 0 2.46959463
## DEFF NaN 1.00000000a.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.SI(N,n,(1/N)*Zd[,"no"])## N y
## Estimation 2396 0.37446198
## Standard Error 0 0.01544825
## CVE 0 4.12545310
## DEFF NaN 1.00000000\[ \begin{align} \widehat{\bar{y}}_{U_{d},\pi}&=\frac{1}{N_d}\frac{N}{n}{\sum}_{s}z_{dk}\\ &=\frac{1}{n}\frac{N}{N_d}{\sum}_{s_d}z_{k} \end{align} \]
\[ \begin{align} V_{MAS}\left(\widehat{\bar{y}}_{U_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\frac{1}{N_d^2}{S}_{z_{d}U}^{2}\\ &=\frac{N^2}{n}\frac{1}{N_d^2}\left(1-\frac{n}{N}\right){S}_{z_{d}U}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{MAS}\left(\widehat{\bar{y}}_{U_{d},\pi}\right)&=\frac{N^2}{n}\left(1-\frac{n}{N}\right)\frac{N^2}{N_d^2}{S}_{z_{d}s}^{2}\\ &=\frac{N^2}{n}\frac{N^2}{N_d^2}\left(1-\frac{n}{N}\right){S}_{z_{d}s}^{2} \end{align} \]
a.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.SI(N,n,(1/N)*Zd[,"yes"]*a.estimar)## N muestra.Income muestra.Employees muestra.Taxes
## Estimation 2396 272.218077 38.710187 7.8192253
## Standard Error 0 9.831566 1.287702 0.5971584
## CVE 0 3.611651 3.326519 7.6370535
## DEFF NaN 1.000000 1.000000 1.0000000a.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.SI(N,n,(1/N)*Zd[,"no"]*a.estimar)## N muestra.Income muestra.Employees muestra.Taxes
## Estimation 2396 156.962697 23.431851 3.9002869
## Standard Error 0 7.832227 1.133919 0.2760593
## CVE 0 4.989865 4.839221 7.0779241
## DEFF NaN 1.000000 1.000000 1.00000009.2.5 El efecto de diseño
\[ \begin{align} {Deff}_{\mathcal{p}(\cdot)}&=\frac{{V}_{\mathcal{p}(\cdot)}\left(\widehat{T}_{y,\cdot}\right)}{{V}_{MAS}\left(\widehat{T}_{y,\pi}\right)}\\ &=\frac{{V}_{\mathcal{p}(\cdot)}\left(\widehat{T}_{y,\cdot}\right)}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}} \end{align} \]
\[ \begin{align} \widehat{Deff}_{\mathcal{p}(\cdot)}&=\frac{\widehat{V}_{\mathcal{p}(\cdot)}\left(\widehat{T}_{y,\cdot}\right)}{\widehat{V}_{MAS}\left(\widehat{T}_{y,\pi}\right)}\\ &=\frac{\widehat{V}_{\mathcal{p}(\cdot)}\left(\widehat{T}_{y,\cdot}\right)}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{ys}^{2}} \end{align} \]
\[ \begin{align} {Deff}_{BER}&=\frac{{V}_{BER}\left(\widehat{T}_{y,\cdot}\right)}{{V}_{MAS}\left(\widehat{T}_{y,\pi}\right)}\\ &=\frac{{V}_{BER}\left(\widehat{t}_{y,\pi}\right)}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\left(\frac{1}{\pi}-1\right){\sum}_{U}y_k^2}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\left(\frac{1}{\frac{n}{N}}-1\right){\sum}_{U}y_k^2}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\left(\frac{N}{n}-1\right)\left[(N-1)S_{yU}^{2}+N\bar{y}_{U}^2\right]}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\left(\frac{N}{n}-1\right)\left[({N}S_{yU}^{2}-S_{yU}^{2}+N\bar{y}_{U}^2\right]}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\left(\frac{N}{n}-1\right)\left[\frac{{N}S_{yU}^{2}}{{N}S_{yU}^{2}}-\frac{S_{yU}^{2}}{{N}S_{yU}^{2}}+\frac{N\bar{y}_{U}^2}{{N}S_{yU}^{2}}\right]{N}S_{yU}^{2}}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\left(\frac{N}{n}-1\right)\left[1-\frac{1}{N}+\frac{1}{{CV}_{yU}^{2}}\right]\frac{N^2}{N}S_{yU}^{2}}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{{N^2}\left(\frac{1}{n}-\frac{1}{N}\right)\left[1-\frac{1}{N}+\frac{1}{{CV}_{yU}^{2}}\right]S_{yU}^{2}}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\left[1-\frac{1}{N}+\frac{1}{{CV}_{yU}^{2}}\right]\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=1-\frac{1}{N}+\frac{1}{{CV}_{yU}^{2}} \end{align} \]
CVyU2 <- sqrt(colMeans(Lucy[,c("Income","Employees","Taxes")]**2)-colMeans(Lucy[,c("Income","Employees","Taxes")])**2)/colMeans(Lucy[,c("Income","Employees","Taxes")]); CVyU2## Income Employees Taxes
## 0.6177919 0.5185117 1.4492537Deff_BER <- 1+(1/N)+(1/(CVyU2**2)); Deff_BER## Income Employees Taxes
## 3.620504 4.719903 1.476532n*Deff_BER## Income Employees Taxes
## 2523.491 3289.772 1029.1439.2.6 Probabilidades de inclusión en unidades de muestreo
\[ \begin{align} {\pi}_{H}&=\mathcal{P}\left(H{\in}s\right)\\ &=1-\mathcal{P}\left(H{\not\in}s\right)\\ &=1-\frac{\binom{M}{0}\binom{N-M}{n}}{\binom{N}{n}}\\ &=1-\frac{\frac{M!}{(M-0)!0!}\frac{(N-M)!}{[(N-M)-n]!n!}}{\frac{N!}{(N-n)!n!}}\\ &=1-\frac{\frac{M!}{M!0!}\frac{(N-M)!}{(N-M-n)!n!}}{\frac{N!}{(N-n)!n!}}\\ &=1-\frac{\frac{(N-M)!}{(N-M-n)!n!}}{\frac{N!}{(N-n)!n!}}\\ &=1-\frac{(N-M)!(N-n)!n!}{(N-M-n)!n!N!}\\ &=1-\frac{(N-M)!(N-n)!}{(N-M-n)!N!}\\ &=1-\frac{(N-M)(N-M-1)\cdots(N-M-n+1)(N-M-n)!(N-n)!}{(N-M-n)!N!}\\ &=1-\frac{(N-M)(N-M-1)\cdots(N-M-n+1)(N-M-n)!(N-n)!}{(N-M-n)!N(N-1)\cdots(N-n+1)(N-n)!}\\ &=1-\frac{(N-M)(N-M-1)\cdots(N-M-n+1)}{N(N-1)\cdots(N-n+1)}\\ \end{align} \]
9.2.6.1 \(M=1\)
\[ \begin{align} {\pi}_{H}&=1-\frac{(N-M)(N-M-1)\cdots(N-M-n+1)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-1)(N-1-1)\cdots(N-1-n+1)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-1)(N-2)\cdots(N-n)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{N-n}{N}\\ &=1-\left[\frac{N}{N}-\frac{n}{N}\right]\\ &=1-\left[1-\frac{n}{N}\right] \end{align} \]
9.2.6.2 \(M=2\) y \(n,N{\rightarrow}+\infty\)
\[ \begin{align} {\pi}_{H}&=1-\frac{(N-M)(N-M-1)\cdots(N-M-n+1)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-2)(N-2-1)\cdots(N-2-n+1)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-2)(N-3)\cdots(N-n-1)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-n)(N-n-1)}{N(N-1)}\\ &=1-\left[\frac{N-n}{N}\right]\left[\frac{N-n-1}{N-1}\right]\\ &=1-\left[\frac{N}{N}-\frac{n}{N}\right]\left[\frac{N-1}{N-1}-\frac{n}{N-1}\right]\\ &=1-\left[1-\frac{n}{N}\right]\left[1-\frac{n}{N-1}\right]\\ &\approx1-\left[1-\frac{n}{N}\right]\left[1-\frac{n}{N}\right]\\ &\approx1-\left[1-\frac{n}{N}\right]^2 \end{align} \]
9.2.6.3 \(M=3\) y \(n,N{\rightarrow}+\infty\)
\[ \begin{align} {\pi}_{H}&=1-\frac{(N-M)(N-M-1)\cdots(N-M-n+1)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-3)(N-3-1)\cdots(N-3-n+1)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-3)(N-4)\cdots(N-n-2)}{N(N-1)\cdots(N-n+1)}\\ &=1-\frac{(N-n)(N-n-1)(N-n-2)}{N(N-1)(N-2)}\\ &=1-\left[\frac{N-n}{N}\right]\left[\frac{N-n-1}{N-1}\right]\left[\frac{N-n-2}{N-2}\right]\\ &=1-\left[\frac{N}{N}-\frac{n}{N}\right]\left[\frac{N-1}{N-1}-\frac{n}{N-1}\right]\left[\frac{N-2}{N-2}-\frac{n}{N-2}\right]\\ &=1-\left[1-\frac{n}{N}\right]\left[1-\frac{n}{N-1}\right]\left[1-\frac{n}{N-2}\right]\\ &\approx1-\left[1-\frac{n}{N}\right]\left[1-\frac{n}{N}\right]\left[1-\frac{n}{N}\right]\\ &\approx1-\left[1-\frac{n}{N}\right]^3 \end{align} \]
9.3 Muestreo aleatorio simple con reemplazo
Diseño de muestreo con reemplazo; se extraen \(m\) muestras, de manera independiente, de tamaño \(1\)
\[m\text{ muestras independientes de tamaño }1\]
\[{\sum}_{U}p_k=1\]
\[p_k\text{: probabilidad de selección}\]
\[ \begin{align} {\forall}_{k{\in}U\&i=1,\ldots,m}\mathcal{P}(k{\in}s_i)&=p_k\\ &=\frac{1}{N} \end{align} \]
El elemento seleccionado es reemplazado en la población y vuelve a ser parte del próximo sorteo aleatorio con la misma probabilidad de selección \(p_k\)
La probabilidad de inclusión \(\pi_k\) no es lo mismo que la probabilidad de selección \(p_k\)
\[\#(S)=n_{k}(S){\leq}m\]
\[ \begin{align} E[n_{k}(S)]&=mp_k\\ &=m\frac{1}{N}\\ &=\frac{m}{N} \end{align} \]
\[ \begin{align} V[n_{k}(S)]&=mp_k(1-p_k)\\ &=m\frac{1}{N}\left[1-\frac{1}{N}\right]\\ &=\frac{m}{N}\left[1-\frac{1}{N}\right] \end{align} \]
\[\pi_k\text{: es la probabilidad de que el elemento sea seleccionado al menos una vez en la muestra}\]
\[ \begin{align} \mathcal{p}(s)&= \begin{cases} \frac{m!}{n_1(s)!n_2(s)!{\cdots}n_N(s)!}{\prod}_{U}({p}_{k})^{n_k(s)} & \text{si }{\sum}_{U}n_k(s)=m\\ 0 & \text{en otro caso} \end{cases}\\&= \begin{cases} \frac{m!}{n_1(s)!n_2(s)!{\cdots}n_N(s)!}{\prod}_{U}\left(\frac{1}{N}\right)^{n_k(s)} & \text{si }{\sum}_{U}n_k(s)=m\\ 0 & \text{en otro caso} \end{cases} \end{align} \]
\[ \begin{align} {\sum}_{s{\in}Q}\mathcal{p}(s)&={\sum}_{s{\in}Q}\frac{m!}{n_1(s)!n_2(s)!{\cdots}n_N(s)!}{\prod}_{U}\left(\frac{1}{N}\right)^{n_k(s)}\\ &={\sum}_{s{\in}Q}\frac{m!}{n_1(s)!n_2(s)!{\cdots}n_N(s)!}\left(\frac{1}{N}\right)^{n_1(s)}\cdots\left(\frac{1}{N}\right)^{n_N(s)}\\ &=\left(\frac{1}{N}+\cdots+\frac{1}{N}\right)^{m}\\ &=\left(\frac{N}{N}\right)^{m}\\ &=1^{m}\\ &=1 \end{align} \]
\[\#(Q)=\binom{N+m-1}{m}\]
\[{\pi}_{k}=1-\left[1-\frac{1}{N}\right]^{m}\]
\[ \begin{align} {\pi}_{kl}&=1-\left[1-\frac{1}{N}\right]^{m}-\left[1-\frac{1}{N}\right]^{m}+\left[1-\frac{1}{N}-\frac{1}{N}\right]^{m}\\ &=1-2\left[1-\frac{1}{N}\right]^{m}+\left[1-\frac{2}{N}\right]^{m} \end{align} \]
9.3.1 Algoritmo de selección
\({0}<{\frac{1}{N}}<{1}\)
\(\forall_{k{\in}U}{\varepsilon}_{k}{\sim}U{\left[0,1\right]}\)
\({\varepsilon}_{k}<{\frac{1}{N}}{\implies}k{\in}s\)
\[\forall_{k{\in}U}\mathcal{P}\left({\varepsilon}_{k}<{\frac{1}{N}}\right)={\frac{1}{N}}{\implies}I_k(S){\sim}Bernoulli\left(\frac{1}{N}\right)\]
9.3.1.1 \(\boldsymbol{m}\) selecciones
Seleccionar un primer elemento
Seleccionar un segundo elemento
Seleccionar un tercer elemento
Seleccionar un \(m\)-ésimo elemento
9.3.1.2 Secuencial
\[{\forall}_{k{\in}U}n_k({s}_{i}){\sim}Binomial\left(m-{\sum}_{i=1}^{k-1}{{n}_{i}},\frac{1}{N-k+1}\right)\]
seleccion <- sample(x=5,size=3,replace=TRUE)
U[seleccion]## [1] "Raul" "Raul" "Jhon"seleccion <- S.WR(N=5,m=3)
U[seleccion]## [1] "Nayibe" "Raul" "Jhon"9.3.2 Estimador de Hansen - Hurtwitz
\[ \begin{align} \widehat{t}_{y,p}&=\frac{1}{m}{\sum}_{U}n_{k}(s)\frac{y_{k}}{p_{k}}\\ &=\frac{1}{m}{\sum}_{U}n_{k}(s)\frac{y_{k}}{\frac{1}{N}}\\ &=\frac{1}{m}{\sum}_{U}n_{k}(s)\frac{\frac{y_{k}}{1}}{\frac{1}{N}}\\ &=\frac{N}{m}{\sum}_{U}n_{k}(s)y_{k} \end{align} \]
\[ \begin{align} E_{MAS_{R}}\left[\widehat{t}_{y,p}\right]&=\frac{N}{m}{\sum}_{U}E_{MAS_{R}}\left[n_{k}(s)\right]y_{k}\\ &=\frac{N}{m}{\sum}_{U}\frac{m}{N}y_{k}\\ &=\frac{N}{m}\frac{m}{N}{\sum}_{U}y_{k}\\ &={\sum}_{U}y_{k}\\ &=t_{y} \end{align} \]
\[ \begin{align} V_{MAS_R}\left[\widehat{t}_{y,p}\right]&=V_{MAS_R}\left[\widehat{t}_{y,p}\right]\\ &=V_{MAS_R}\left[\frac{1}{m}{\sum}_{i=1}^{m}Z_{i}\right]\\ &=\frac{1}{m^{2}}{\sum}_{i=1}^{m}V_{MAS_R}\left[Z_{i}\right]\\ &=\frac{1}{m^{2}}{\sum}_{i=1}^{m}{\sum}_{U}\left[\frac{y_{k}}{p_{k}}-t_{y}\right]^{2}p_{k}\\ &=\frac{1}{m^{2}}{\sum}_{i=1}^{m}{\sum}_{U}\left[\frac{y_{k}}{\frac{1}{N}}-t_{y}\right]^{2}\frac{1}{N}\\ &=\frac{1}{m^{2}}{\sum}_{i=1}^{m}\frac{1}{N}{\sum}_{U}\left[\frac{\frac{y_{k}}{1}}{\frac{1}{N}}-t_{y}\right]^{2}\\ &=\frac{1}{m^{2}}{\sum}_{i=1}^{m}\frac{1}{N}{\sum}_{U}\left(Ny_{k}-t_{y}\right)^{2}\\ &=\frac{1}{m^{2}}{\sum}_{i=1}^{m}\frac{1}{N}{\sum}_{U}\left[Ny_{k}-\frac{N}{N}t_{y}\right]^{2}\\ &=\frac{1}{m^{2}}{\sum}_{i=1}^{m}\frac{N^2}{N}{\sum}_{U}\left[y_{k}-\frac{t_{y}}{N}\right]^{2}\\ &=\frac{1}{m^{2}}{m}\frac{N^2}{N}{\sum}_{U}\left(y_{k}-\bar{y}_{U}\right)^{2}\\ &=\frac{1}{m}{N}\left[\frac{N-1}{N-1}\right]{\sum}_{U}\left(y_{k}-\bar{y}_{U}\right)^{2}\\ &=\frac{1}{m}{N}\left(N-1\right)\frac{1}{N-1}{\sum}_{U}\left(y_{k}-\bar{y}_{U}\right)^{2}\\ &=\frac{N\left(N-1\right)}{m}S_{yU}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{MAS_R}\left[\widehat{t}_{y,p}\right]&=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}n_{k}(s)\left[\frac{y_{k}}{p_{k}}-\widehat{t}_{y,p}\right]^{2}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}n_{k}(s)\left[\frac{y_{k}}{\frac{1}{N}}-\widehat{t}_{y,p}\right]^{2}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}n_{k}(s)\left[\frac{\frac{y_{k}}{1}}{\frac{1}{N}}-\widehat{t}_{y,p}\right]^{2}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}n_{k}(s)\left[{N}y_{k}-\widehat{t}_{y,p}\right]^{2}\\ \end{align} \]
\[ \begin{align} E_{MAS_R}\left\{\widehat{V}_{MAS_R}\left[\widehat{t}_{y,p}\right]\right\}&=E_{MAS_R}\left\{\frac{1}{m}\frac{1}{m-1}{\sum}_{U}n_{k}(s)\left[{N}y_{k}-\widehat{t}_{y,p}\right]^{2}\right\}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}E_{MAS_R}\left\{n_{k}(s)\left[{N}y_{k}-\widehat{t}_{y,p}\right]^{2}\right\}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}E_{MAS_R}\left\{n_{k}(s)\left[{N}y_{k}-t_{y}-\widehat{t}_{y,p}+t_{y}\right]^{2}\right\}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}E_{MAS_R}\left\{n_{k}(s)\left[\left({N}y_{k}-t_{y}\right)-\left(\widehat{t}_{y,p}-t_{y}\right)\right]^{2}\right\}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}E_{MAS_R}\left\{n_{k}(s)\left[\left({N}y_{k}-t_{y}\right)^{2}-2\left({N}y_{k}-t_{y}\right)\left(\widehat{t}_{y,p}-t_{y}\right)+\left(\widehat{t}_{y,p}-t_{y}\right)^{2}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}E_{MAS_R}\left\{n_{k}(s)\left[\left({N}y_{k}-t_{y}\right)^{2}-2\frac{m}{m}\left({N}y_{k}-t_{y}\right)\left(\widehat{t}_{y,p}-t_{y}\right)+\left(\widehat{t}_{y,p}-t_{y}\right)^{2}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}{\sum}_{U}E_{MAS_R}\left\{n_{k}(s)\left[\left({N}y_{k}-t_{y}\right)^{2}-2{m}\left(\frac{N}{m}y_{k}-\frac{t_{y}}{m}\right)\left(\widehat{t}_{y,p}-t_{y}\right)+\left(\widehat{t}_{y,p}-t_{y}\right)^{2}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{\sum}_{U}E_{MAS_R}\left[n_{k}(s)\left({N}y_{k}-t_{y}\right)^{2}\right]-{\sum}_{U}E_{MAS_R}\left[n_{k}(s)\left(\widehat{t}_{y,p}-t_{y}\right)^{2}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{\sum}_{U}\frac{m}{N}\left({N}y_{k}-t_{y}\right)^{2}-{\sum}_{U}\frac{m}{N}E_{MAS_R}\left[\left(\widehat{t}_{y,p}-t_{y}\right)^{2}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{m}{\sum}_{U}\frac{1}{N}\left({N}y_{k}-t_{y}\right)^{2}-{N}\frac{m}{N}E_{MAS_R}\left[\left(\widehat{t}_{y,p}-t_{y}\right)^{2}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{m}{\sum}_{U}\frac{N^2}{N}\left(y_{k}-\frac{t_{y}}{N}\right)^{2}-{N}\frac{m}{N}E_{MAS_R}\left[\left(\widehat{t}_{y,p}-t_{y}\right)^{2}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{m^2}\frac{1}{m}{N}{\sum}_{U}\left(y_{k}-\bar{y}_{U}\right)^{2}-{m}V_{MAS_R}\left[\widehat{t}_{y,p}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{m^2}\frac{N}{m}(N-1)\frac{1}{N-1}{\sum}_{U}\left(y_{k}-\bar{y}_{U}\right)^{2}-{m}V_{MAS_R}\left[\widehat{t}_{y,p}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{m^2}\frac{N(N-1)}{m}S_{yU}^{2}-{m}V_{MAS_R}\left[\widehat{t}_{y,p}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{m^2}V_{MAS_R}\left[\widehat{t}_{y,p}\right]-{m}V_{MAS_R}\left[\widehat{t}_{y,p}\right]\right\}\\ &=\frac{1}{m}\frac{1}{m-1}\left\{{m}^{2}-m\right\}V_{MAS_R}\left[\widehat{t}_{y,p}\right]\\ &=V_{MAS_R}\left[\widehat{t}_{y,p}\right] \end{align} \]
9.3.2.1 Estimación del total y tamaño de muestra
9.3.2.1.1 \(m=697\)
attach(Lucy)seleccion <- S.WR(N,n)
muestra <- Lucy[seleccion,]
attach(muestra);muestra## ID Ubication Level Zone Income Employees Taxes SPAM
## 5 AB005 c1k5 Small A 391 91 7.0 yes
## 6 AB006 c1k6 Small A 296 89 3.0 no
## 9 AB009 c1k9 Small A 350 84 5.0 yes
## 9.1 AB009 c1k9 Small A 350 84 5.0 yes
## 15 AB015 c1k15 Small A 411 36 7.0 no
## 17 AB017 c1k17 Small A 311 34 4.0 no
## 18 AB018 c1k18 Small A 342 40 5.0 yes
## 19 AB019 c1k19 Small A 342 60 5.0 yes
## 19.1 AB019 c1k19 Small A 342 60 5.0 yes
## 21 AB021 c1k21 Small A 425 49 8.0 yes
## 29 AB029 c1k29 Small A 310 94 4.0 yes
## 33 AB033 c1k33 Small A 334 72 5.0 yes
## 34 AB034 c1k34 Small A 350 80 5.0 no
## 35 AB035 c1k35 Small A 381 94 6.0 no
## 36 AB036 c1k36 Small A 343 52 5.0 yes
## 36.1 AB036 c1k36 Small A 343 52 5.0 yes
## 41 AB041 c1k41 Small A 340 20 5.0 yes
## 43 AB043 c1k43 Small A 440 22 8.0 yes
## 44 AB044 c1k44 Small A 337 44 5.0 no
## 45 AB045 c1k45 Small A 365 53 6.0 yes
## 49 AB050 c1k49 Small A 334 16 5.0 no
## 61 AB064 c1k61 Small A 350 76 5.0 no
## 65 AB068 c1k65 Small A 360 61 5.0 no
## 65.1 AB068 c1k65 Small A 360 61 5.0 no
## 72 AB077 c1k72 Small A 363 81 5.0 yes
## 77 AB082 c1k77 Small A 410 24 7.0 yes
## 77.1 AB082 c1k77 Small A 410 24 7.0 yes
## 82 AB087 c1k82 Small A 238 83 2.0 yes
## 84 AB089 c1k84 Small A 481 65 10.5 yes
## 85 AB092 c1k85 Small A 490 66 10.5 yes
## 94 AB1135 c1k94 Small A 470 32 10.0 yes
## 95 AB1151 c1k95 Small A 490 98 10.5 yes
## 98 AB1193 c1k98 Small B 490 38 10.5 yes
## 105 AB1229 c2k6 Small B 388 91 6.0 no
## 112 AB126 c2k13 Small B 65 69 0.5 yes
## 118 AB1305 c2k19 Small B 436 77 8.0 yes
## 120 AB132 c2k21 Small B 125 34 0.5 no
## 121 AB133 c2k22 Small B 28 73 0.5 yes
## 128 AB1338 c2k29 Small B 282 69 3.0 yes
## 132 AB1341 c2k33 Small B 304 18 4.0 no
## 133 AB1342 c2k34 Small B 290 93 3.0 yes
## 134 AB1343 c2k35 Small B 340 36 5.0 no
## 137 AB1346 c2k38 Small B 340 96 5.0 yes
## 142 AB1350 c2k43 Small B 280 69 3.0 no
## 143 AB1351 c2k44 Small B 245 39 2.0 no
## 147 AB1355 c2k48 Small B 266 16 3.0 no
## 151 AB1359 c2k52 Small B 370 70 6.0 yes
## 167 AB1373 c2k68 Small B 355 33 5.0 yes
## 177 AB1382 c2k78 Small B 319 55 4.0 yes
## 187 AB1391 c2k88 Small B 290 77 3.0 no
## 191 AB1395 c2k92 Small B 339 48 5.0 no
## 193 AB1397 c2k94 Small B 350 48 5.0 yes
## 194 AB1398 c2k95 Small B 330 39 4.0 yes
## 199 AB1402 c3k1 Small B 280 61 3.0 yes
## 201 AB1404 c3k3 Small B 334 60 5.0 yes
## 202 AB1405 c3k4 Small B 290 57 3.0 yes
## 204 AB1407 c3k6 Small B 302 18 4.0 yes
## 207 AB141 c3k9 Small B 128 66 0.5 yes
## 210 AB1412 c3k12 Small B 314 58 4.0 yes
## 221 AB1422 c3k23 Small B 295 57 3.0 no
## 224 AB1425 c3k26 Small B 287 25 3.0 yes
## 234 AB1434 c3k36 Small B 257 44 2.0 yes
## 234.1 AB1434 c3k36 Small B 257 44 2.0 yes
## 239 AB1439 c3k41 Small B 297 57 3.0 no
## 239.1 AB1439 c3k41 Small B 297 57 3.0 no
## 241 AB1440 c3k43 Small B 270 28 3.0 yes
## 242 AB1441 c3k44 Small B 351 29 5.0 yes
## 245 AB1444 c3k47 Small B 390 19 7.0 yes
## 247 AB1446 c3k49 Small B 333 48 5.0 yes
## 252 AB1450 c3k54 Small B 334 80 5.0 no
## 253 AB1451 c3k55 Small B 232 47 2.0 no
## 257 AB1455 c3k59 Small B 270 76 3.0 yes
## 261 AB1459 c3k63 Small B 204 65 1.0 yes
## 265 AB1462 c3k67 Small B 330 32 4.0 yes
## 267 AB1464 c3k69 Small B 314 18 4.0 no
## 282 AB1478 c3k84 Small B 267 88 3.0 no
## 284 AB148 c3k86 Small B 91 29 0.5 no
## 284.1 AB148 c3k86 Small B 91 29 0.5 no
## 293 AB1488 c3k95 Small B 209 30 1.0 yes
## 296 AB1490 c3k98 Small B 245 67 2.0 no
## 297 AB1491 c3k99 Small B 296 13 3.0 yes
## 300 AB1494 c4k3 Small B 215 30 2.0 yes
## 301 AB1495 c4k4 Small B 240 31 2.0 yes
## 302 AB1496 c4k5 Small B 202 73 1.0 no
## 302.1 AB1496 c4k5 Small B 202 73 1.0 no
## 309 AB1502 c4k12 Small B 253 44 2.0 yes
## 318 AB1510 c4k21 Small B 260 12 2.0 yes
## 321 AB1513 c4k24 Small B 196 45 1.0 yes
## 323 AB1515 c4k26 Small B 241 87 2.0 yes
## 325 AB1517 c4k28 Small B 208 22 1.0 no
## 330 AB1521 c4k33 Small B 344 84 5.0 yes
## 331 AB1522 c4k34 Small B 201 85 1.0 yes
## 338 AB1529 c4k41 Small B 266 20 3.0 yes
## 340 AB1530 c4k43 Small B 328 75 4.0 yes
## 342 AB1532 c4k45 Small B 271 80 3.0 yes
## 348 AB1538 c4k51 Small B 280 13 3.0 yes
## 349 AB1539 c4k52 Small B 314 54 4.0 no
## 352 AB1541 c4k55 Small B 310 54 4.0 yes
## 355 AB1544 c4k58 Small B 245 63 2.0 yes
## 364 AB1552 c4k67 Small B 227 6 2.0 yes
## 368 AB1556 c4k71 Small B 313 86 4.0 yes
## 368.1 AB1556 c4k71 Small B 313 86 4.0 yes
## 368.2 AB1556 c4k71 Small B 313 86 4.0 yes
## 369 AB1557 c4k72 Small B 290 89 3.0 yes
## 370 AB1558 c4k73 Small B 206 10 1.0 yes
## 371 AB1559 c4k74 Small B 260 76 2.0 no
## 373 AB1560 c4k76 Small B 270 72 3.0 yes
## 374 AB1561 c4k77 Small B 317 67 4.0 yes
## 374.1 AB1561 c4k77 Small B 317 67 4.0 yes
## 375 AB1562 c4k78 Small B 296 45 3.0 yes
## 378 AB1565 c4k81 Small B 206 46 1.0 no
## 382 AB1569 c4k85 Small B 345 68 5.0 yes
## 389 AB1575 c4k92 Small B 273 16 3.0 yes
## 391 AB1577 c4k94 Small B 313 46 4.0 yes
## 391.1 AB1577 c4k94 Small B 313 46 4.0 yes
## 395 AB1580 c4k98 Small B 299 85 3.0 no
## 396 AB1581 c4k99 Small B 235 55 2.0 no
## 396.1 AB1581 c4k99 Small B 235 55 2.0 no
## 397 AB1582 c5k1 Small B 328 39 4.0 no
## 400 AB1585 c5k4 Small B 300 54 3.0 yes
## 405 AB159 c5k9 Small B 65 73 0.5 yes
## 409 AB1593 c5k13 Small B 293 81 3.0 yes
## 410 AB1594 c5k14 Small B 264 44 3.0 no
## 414 AB1598 c5k18 Small B 292 89 3.0 no
## 415 AB1599 c5k19 Small B 268 44 3.0 no
## 416 AB160 c5k20 Small B 179 16 1.0 no
## 432 AB1614 c5k36 Small B 276 16 3.0 yes
## 442 AB1623 c5k46 Small B 341 52 5.0 yes
## 444 AB1625 c5k48 Small B 303 54 4.0 yes
## 446 AB1627 c5k50 Small B 295 29 3.0 yes
## 450 AB1630 c5k54 Small B 236 31 2.0 yes
## 451 AB1631 c5k55 Small B 226 18 2.0 no
## 458 AB1638 c5k62 Small B 264 40 3.0 no
## 463 AB1642 c5k67 Small B 235 75 2.0 no
## 467 AB1646 c5k71 Small B 334 76 5.0 no
## 474 AB1652 c5k78 Small B 369 97 6.0 yes
## 489 AB1666 c5k93 Small B 275 44 3.0 no
## 491 AB1668 c5k95 Small B 336 80 5.0 yes
## 501 AB1677 c6k6 Small B 248 71 2.0 no
## 504 AB168 c6k9 Small B 87 9 0.5 yes
## 505 AB1680 c6k10 Small B 235 91 2.0 yes
## 511 AB1686 c6k16 Small B 269 36 3.0 yes
## 512 AB1687 c6k17 Small B 315 63 4.0 yes
## 518 AB1692 c6k23 Small B 130 14 0.5 yes
## 522 AB1696 c6k27 Small B 212 14 1.0 no
## 526 AB170 c6k31 Small B 75 73 0.5 yes
## 528 AB1701 c6k33 Small B 195 37 1.0 no
## 535 AB1708 c6k40 Small B 154 7 1.0 yes
## 536 AB1709 c6k41 Small B 220 82 2.0 no
## 538 AB1710 c6k43 Small B 1 45 0.5 yes
## 539 AB1711 c6k44 Small B 180 12 1.0 yes
## 539.1 AB1711 c6k44 Small B 180 12 1.0 yes
## 542 AB1714 c6k47 Small B 130 50 0.5 no
## 544 AB1716 c6k49 Small B 125 74 0.5 no
## 544.1 AB1716 c6k49 Small B 125 74 0.5 no
## 558 AB1729 c6k63 Small B 106 53 0.5 yes
## 558.1 AB1729 c6k63 Small B 106 53 0.5 yes
## 559 AB173 c6k64 Small B 144 15 0.5 yes
## 566 AB1736 c6k71 Small B 130 66 0.5 no
## 574 AB1743 c6k79 Small B 196 89 1.0 yes
## 582 AB1750 c6k87 Small B 164 72 1.0 yes
## 588 AB1756 c6k93 Small B 141 27 0.5 yes
## 592 AB176 c6k97 Small B 83 85 0.5 yes
## 601 AB1768 c7k7 Small B 190 25 1.0 yes
## 612 AB1778 c7k18 Small B 134 31 0.5 yes
## 618 AB1783 c7k24 Small B 220 74 2.0 yes
## 635 AB1799 c7k41 Small B 130 38 0.5 yes
## 641 AB1804 c7k47 Small B 136 83 0.5 no
## 647 AB181 c7k53 Small B 194 85 1.0 yes
## 648 AB1810 c7k54 Small B 156 23 1.0 yes
## 652 AB1814 c7k58 Small B 86 77 0.5 no
## 653 AB1815 c7k59 Small B 162 56 1.0 yes
## 653.1 AB1815 c7k59 Small B 162 56 1.0 yes
## 655 AB1817 c7k61 Small B 191 25 1.0 yes
## 657 AB1819 c7k63 Small B 97 25 0.5 yes
## 659 AB1820 c7k65 Small B 131 74 0.5 no
## 660 AB1821 c7k66 Small B 80 41 0.5 yes
## 660.1 AB1821 c7k66 Small B 80 41 0.5 yes
## 664 AB1825 c7k70 Small B 181 64 1.0 yes
## 667 AB1828 c7k73 Small B 94 41 0.5 no
## 670 AB1830 c7k76 Small B 117 38 0.5 no
## 673 AB1833 c7k79 Small B 152 83 1.0 yes
## 675 AB1835 c7k81 Small B 131 82 0.5 yes
## 676 AB1836 c7k82 Small B 144 11 0.5 yes
## 677 AB1837 c7k83 Small B 170 60 1.0 no
## 678 AB1838 c7k84 Small B 157 59 1.0 yes
## 689 AB1848 c7k95 Small B 179 24 1.0 yes
## 695 AB1853 c8k2 Small C 190 21 1.0 yes
## 700 AB1858 c8k7 Small C 176 20 1.0 no
## 702 AB186 c8k9 Small C 67 77 0.5 no
## 704 AB1861 c8k11 Small C 120 22 0.5 yes
## 709 AB1866 c8k16 Small C 196 57 1.0 yes
## 712 AB1869 c8k19 Small C 172 56 1.0 yes
## 713 AB187 c8k20 Small C 98 65 0.5 yes
## 720 AB1876 c8k27 Small C 135 51 0.5 yes
## 720.1 AB1876 c8k27 Small C 135 51 0.5 yes
## 730 AB1885 c8k37 Small C 380 18 6.0 yes
## 731 AB1886 c8k38 Small C 335 64 5.0 no
## 737 AB1891 c8k44 Small C 470 44 10.0 yes
## 744 AB1898 c8k51 Small C 370 98 6.0 yes
## 745 AB1899 c8k52 Small C 390 31 6.0 no
## 745.1 AB1899 c8k52 Small C 390 31 6.0 no
## 748 AB1901 c8k55 Small C 384 50 6.0 no
## 752 AB1906 c8k59 Small C 460 43 9.0 no
## 756 AB1910 c8k63 Small C 350 92 5.0 yes
## 756.1 AB1910 c8k63 Small C 350 92 5.0 yes
## 764 AB1918 c8k71 Small C 390 39 7.0 yes
## 765 AB1919 c8k72 Small C 410 36 7.0 yes
## 769 AB1922 c8k76 Small C 410 36 7.0 no
## 769.1 AB1922 c8k76 Small C 410 36 7.0 no
## 770 AB1923 c8k77 Small C 450 58 9.0 yes
## 770.1 AB1923 c8k77 Small C 450 58 9.0 yes
## 773 AB1927 c8k80 Small C 420 48 7.0 yes
## 777 AB1930 c8k84 Small C 380 34 6.0 yes
## 778 AB1931 c8k85 Small C 416 52 7.0 yes
## 781 AB1934 c8k88 Small C 470 52 10.0 yes
## 782 AB1936 c8k89 Small C 310 18 4.0 no
## 784 AB1938 c8k91 Small C 410 36 7.0 yes
## 786 AB194 c8k93 Small C 153 27 1.0 no
## 786.1 AB194 c8k93 Small C 153 27 1.0 no
## 791 AB1946 c8k98 Small C 390 59 7.0 yes
## 791.1 AB1946 c8k98 Small C 390 59 7.0 yes
## 795 AB195 c9k3 Small C 198 49 1.0 yes
## 801 AB1955 c9k9 Small C 440 82 8.0 no
## 802 AB1956 c9k10 Small C 470 76 10.0 yes
## 806 AB1960 c9k14 Small C 312 74 4.0 yes
## 811 AB1965 c9k19 Small C 475 89 10.0 yes
## 822 AB1975 c9k30 Small C 420 36 8.0 yes
## 831 AB1983 c9k39 Small C 440 94 8.0 yes
## 832 AB1984 c9k40 Small C 480 25 10.0 yes
## 833 AB1985 c9k41 Small C 300 14 3.0 yes
## 838 AB1990 c9k46 Small C 378 62 6.0 yes
## 838.1 AB1990 c9k46 Small C 378 62 6.0 yes
## 840 AB1992 c9k48 Small C 430 37 8.0 no
## 841 AB1993 c9k49 Small C 410 24 7.0 no
## 842 AB1994 c9k50 Small C 440 94 8.0 no
## 845 AB1997 c9k53 Small C 400 95 7.0 yes
## 853 AB2005 c9k61 Small C 390 47 6.0 yes
## 857 AB2009 c9k65 Small C 491 74 10.5 yes
## 861 AB2016 c9k69 Small C 436 97 8.0 yes
## 862 AB2017 c9k70 Small C 363 97 5.0 no
## 862.1 AB2017 c9k70 Small C 363 97 5.0 no
## 862.2 AB2017 c9k70 Small C 363 97 5.0 no
## 865 AB2021 c9k73 Small C 324 23 4.0 no
## 866 AB2022 c9k74 Small C 370 54 6.0 no
## 867 AB2023 c9k75 Small C 469 20 10.0 no
## 875 AB2032 c9k83 Small C 400 19 7.0 yes
## 877 AB2034 c9k85 Small C 434 53 8.0 yes
## 879 AB2036 c9k87 Small C 490 54 10.5 yes
## 881 AB2038 c9k89 Small C 460 99 9.0 yes
## 883 AB204 c9k91 Small C 91 65 0.5 yes
## 884 AB2040 c9k92 Small C 430 81 8.0 yes
## 886 AB2043 c9k94 Small C 400 79 7.0 yes
## 887 AB2044 c9k95 Small C 404 87 7.0 yes
## 889 AB2047 c9k97 Small C 385 94 6.0 yes
## 890 AB2049 c9k98 Small C 480 85 10.0 yes
## 890.1 AB2049 c9k98 Small C 480 85 10.0 yes
## 897 AB2058 c10k6 Small C 480 93 10.0 no
## 899 AB206 c10k8 Small C 110 62 0.5 no
## 899.1 AB206 c10k8 Small C 110 62 0.5 no
## 901 AB2062 c10k10 Small C 333 76 5.0 no
## 903 AB2064 c10k12 Small C 420 76 8.0 yes
## 907 AB2069 c10k16 Small C 390 31 7.0 no
## 908 AB207 c10k17 Small C 171 12 1.0 yes
## 910 AB2071 c10k19 Small C 375 30 6.0 no
## 912 AB2073 c10k21 Small C 420 76 8.0 no
## 913 AB2074 c10k22 Small C 332 28 4.0 yes
## 921 AB2082 c10k30 Small C 480 41 10.0 yes
## 921.1 AB2082 c10k30 Small C 480 41 10.0 yes
## 939 AB2103 c10k48 Small C 308 82 4.0 no
## 947 AB2114 c10k56 Small C 361 37 5.0 yes
## 947.1 AB2114 c10k56 Small C 361 37 5.0 yes
## 951 AB2118 c10k60 Small C 480 85 10.0 no
## 954 AB2120 c10k63 Small C 350 20 5.0 yes
## 956 AB2122 c10k65 Small C 420 28 7.0 no
## 971 AB2139 c10k80 Small C 332 20 4.0 no
## 973 AB2141 c10k82 Small C 480 21 10.0 yes
## 974 AB2142 c10k83 Small C 390 19 7.0 yes
## 974.1 AB2142 c10k83 Small C 390 19 7.0 yes
## 978 AB2146 c10k87 Small C 476 81 10.0 yes
## 979 AB2147 c10k88 Small C 487 105 10.5 yes
## 985 AB2152 c10k94 Small C 452 63 9.0 no
## 987 AB2155 c10k96 Small C 480 41 10.0 no
## 987.1 AB2155 c10k96 Small C 480 41 10.0 no
## 1002 AB2169 c11k12 Small C 260 68 2.0 no
## 1010 AB2178 c11k20 Small C 494 38 10.5 no
## 1016 AB2186 c11k26 Small C 345 40 5.0 yes
## 1024 AB2195 c11k34 Small C 445 38 9.0 yes
## 1032 AB2203 c11k42 Small C 421 85 8.0 yes
## 1038 AB221 c11k48 Small C 20 41 0.5 no
## 1046 AB2218 c11k56 Small C 365 85 6.0 no
## 1048 AB222 c11k58 Small C 110 78 0.5 no
## 1056 AB223 c11k66 Small C 89 65 0.5 yes
## 1060 AB2233 c11k70 Small C 380 26 6.0 yes
## 1067 AB2240 c11k77 Small C 374 18 6.0 yes
## 1067.1 AB2240 c11k77 Small C 374 18 6.0 yes
## 1071 AB2244 c11k81 Small C 491 94 10.5 no
## 1074 AB2247 c11k84 Small C 320 35 4.0 yes
## 1075 AB2248 c11k85 Small C 411 52 7.0 yes
## 1083 AB2258 c11k93 Small C 394 87 7.0 yes
## 1084 AB2259 c11k94 Small C 463 24 10.0 yes
## 1085 AB226 c11k95 Small C 101 73 0.5 yes
## 1092 AB2268 c12k3 Small C 385 74 6.0 no
## 1094 AB227 c12k5 Small C 122 54 0.5 yes
## 1097 AB2273 c12k8 Small C 357 21 5.0 no
## 1098 AB2274 c12k9 Small C 443 74 8.0 no
## 1100 AB2276 c12k11 Small C 410 88 7.0 yes
## 1100.1 AB2276 c12k11 Small C 410 88 7.0 yes
## 1102 AB2278 c12k13 Small C 374 14 6.0 yes
## 1106 AB2282 c12k17 Small C 362 45 5.0 yes
## 1117 AB2292 c12k28 Small C 460 79 9.0 yes
## 1121 AB2296 c12k32 Small C 461 44 9.0 no
## 1121.1 AB2296 c12k32 Small C 461 44 9.0 no
## 1125 AB230 c12k36 Small C 158 15 1.0 yes
## 1131 AB2306 c12k42 Small C 350 25 5.0 yes
## 1134 AB2310 c12k45 Small C 324 31 4.0 no
## 1134.1 AB2310 c12k45 Small C 324 31 4.0 no
## 1136 AB2312 c12k47 Small C 424 61 8.0 yes
## 1140 AB2316 c12k51 Small C 392 47 7.0 yes
## 1142 AB2318 c12k53 Small C 414 28 7.0 yes
## 1144 AB232 c12k55 Small C 87 9 0.5 yes
## 1147 AB2322 c12k58 Small C 416 80 7.0 no
## 1151 AB2326 c12k62 Small C 424 45 8.0 no
## 1155 AB2330 c12k66 Small C 354 25 5.0 yes
## 1162 AB2339 c12k73 Small C 425 21 8.0 yes
## 1165 AB2342 c12k76 Small C 409 52 7.0 no
## 1166 AB2343 c12k77 Small C 455 67 9.0 no
## 1167 AB2344 c12k78 Small C 388 95 6.0 yes
## 1174 AB2350 c12k85 Small C 373 26 6.0 yes
## 1179 AB2356 c12k90 Small C 436 29 8.0 no
## 1179.1 AB2356 c12k90 Small C 436 29 8.0 no
## 1181 AB2358 c12k92 Small C 348 56 5.0 yes
## 1181.1 AB2358 c12k92 Small C 348 56 5.0 yes
## 1182 AB2359 c12k93 Small C 367 53 6.0 yes
## 1192 AB2368 c13k4 Small C 438 61 8.0 yes
## 1196 AB2371 c13k8 Small C 400 55 7.0 yes
## 1198 AB2373 c13k10 Small C 452 79 9.0 yes
## 1200 AB2375 c13k12 Small C 489 101 10.5 yes
## 1201 AB2377 c13k13 Small C 365 81 6.0 yes
## 1203 AB2380 c13k15 Small C 421 21 8.0 no
## 1203.1 AB2380 c13k15 Small C 421 21 8.0 no
## 1205 AB2383 c13k17 Small C 484 69 10.5 no
## 1207 AB2385 c13k19 Small C 375 46 6.0 yes
## 1209 AB2387 c13k21 Small C 405 52 7.0 no
## 1216 AB2393 c13k28 Small C 410 60 7.0 yes
## 1222 AB243 c13k34 Small C 93 65 0.5 no
## 1227 AB248 c13k39 Small C 117 18 0.5 yes
## 1232 AB253 c13k44 Small C 154 75 1.0 yes
## 1236 AB257 c13k48 Small C 97 25 0.5 no
## 1242 AB263 c13k54 Small C 177 48 1.0 no
## 1245 AB266 c13k57 Small C 93 5 0.5 no
## 1251 AB272 c13k63 Small C 112 70 0.5 yes
## 1251.1 AB272 c13k63 Small C 112 70 0.5 yes
## 1255 AB276 c13k67 Small C 119 38 0.5 yes
## 1266 AB287 c13k78 Small C 127 42 0.5 no
## 1277 AB298 c13k89 Small C 153 7 1.0 no
## 1283 AB304 c13k95 Small C 149 7 0.5 yes
## 1283.1 AB304 c13k95 Small C 149 7 0.5 yes
## 1285 AB306 c13k97 Small C 148 79 0.5 no
## 1286 AB307 c13k98 Small C 101 77 0.5 yes
## 1287 AB308 c13k99 Small C 120 46 0.5 yes
## 1292 AB313 c14k5 Small C 196 77 1.0 no
## 1293 AB314 c14k6 Small C 152 79 1.0 yes
## 1296 AB317 c14k9 Small C 96 69 0.5 yes
## 1297 AB318 c14k10 Small C 103 33 0.5 no
## 1298 AB319 c14k11 Small C 98 25 0.5 yes
## 1298.1 AB319 c14k11 Small C 98 25 0.5 yes
## 1300 AB321 c14k13 Small C 133 35 0.5 yes
## 1308 AB329 c14k21 Small C 168 8 1.0 no
## 1310 AB331 c14k23 Small C 169 28 1.0 yes
## 1315 AB336 c14k28 Small C 131 22 0.5 no
## 1316 AB337 c14k29 Small C 131 74 0.5 no
## 1316.1 AB337 c14k29 Small C 131 74 0.5 no
## 1323 AB344 c14k36 Small C 127 14 0.5 yes
## 1327 AB348 c14k40 Small C 119 78 0.5 yes
## 1332 AB353 c14k45 Small C 128 62 0.5 yes
## 1333 AB354 c14k46 Small C 174 32 1.0 yes
## 1333.1 AB354 c14k46 Small C 174 32 1.0 yes
## 1335 AB356 c14k48 Small C 209 90 1.0 no
## 1340 AB361 c14k53 Small C 145 27 0.5 yes
## 1341 AB362 c14k54 Small C 191 45 1.0 yes
## 1341.1 AB362 c14k54 Small C 191 45 1.0 yes
## 1344 AB365 c14k57 Small C 143 55 0.5 no
## 1345 AB366 c14k58 Small C 78 13 0.5 yes
## 1350 AB371 c14k63 Small C 120 10 0.5 no
## 1366 AB387 c14k79 Small C 64 37 0.5 yes
## 1367 AB388 c14k80 Small C 96 29 0.5 yes
## 1373 AB394 c14k86 Small C 132 82 0.5 yes
## 1375 AB396 c14k88 Small C 99 65 0.5 yes
## 1380 AB401 c14k93 Small C 188 73 1.0 yes
## 1380.1 AB401 c14k93 Small C 188 73 1.0 yes
## 1385 AB406 c14k98 Small C 119 54 0.5 no
## 1387 AB408 c15k1 Small C 119 66 0.5 yes
## 1389 AB410 c15k3 Small C 165 76 1.0 yes
## 1391 AB412 c15k5 Small C 76 85 0.5 yes
## 1403 AB424 c15k17 Small C 230 39 2.0 yes
## 1404 AB425 c15k18 Small C 305 66 4.0 yes
## 1408 AB429 c15k22 Small C 270 60 3.0 no
## 1409 AB430 c15k23 Small C 276 48 3.0 yes
## 1422 AB443 c15k36 Small C 249 67 2.0 no
## 1426 AB447 c15k40 Small C 250 91 2.0 yes
## 1426.1 AB447 c15k40 Small C 250 91 2.0 yes
## 1429 AB450 c15k43 Small C 292 29 3.0 yes
## 1437 AB458 c15k51 Small C 236 43 2.0 yes
## 1447 AB468 c15k61 Small C 272 40 3.0 no
## 1452 AB473 c15k66 Small C 247 91 2.0 no
## 1454 AB475 c15k68 Small C 235 59 2.0 no
## 1454.1 AB475 c15k68 Small C 235 59 2.0 no
## 1458 AB479 c15k72 Small C 300 66 3.0 yes
## 1464 AB485 c15k78 Small C 237 23 2.0 yes
## 1466 AB487 c15k80 Small C 259 28 2.0 yes
## 1470 AB491 c15k84 Small C 219 10 2.0 yes
## 1476 AB497 c15k90 Small C 239 31 2.0 yes
## 1479 AB500 c15k93 Small C 278 17 3.0 no
## 1481 AB502 c15k95 Small C 268 52 3.0 no
## 1484 AB505 c15k98 Small C 219 6 2.0 no
## 1485 AB506 c15k99 Small C 258 52 2.0 yes
## 1488 AB509 c16k3 Small C 226 38 2.0 yes
## 1489 AB510 c16k4 Small C 213 50 1.0 yes
## 1501 AB522 c16k16 Small C 248 83 2.0 yes
## 1502 AB523 c16k17 Small C 299 89 3.0 no
## 1506 AB527 c16k21 Small C 179 40 1.0 yes
## 1507 AB528 c16k22 Small C 214 34 1.0 no
## 1509 AB530 c16k24 Small C 328 15 4.0 no
## 1511 AB532 c16k26 Small C 245 39 2.0 no
## 1513 AB534 c16k28 Small C 253 60 2.0 yes
## 1516 AB537 c16k31 Small C 242 23 2.0 no
## 1517 AB538 c16k32 Small C 218 22 2.0 no
## 1520 AB541 c16k35 Small C 227 34 2.0 yes
## 1523 AB544 c16k38 Small C 341 16 5.0 yes
## 1524 AB545 c16k39 Small C 296 41 3.0 yes
## 1528 AB549 c16k43 Small C 234 11 2.0 yes
## 1531 AB552 c16k46 Small C 243 47 2.0 no
## 1534 AB555 c16k49 Small C 279 85 3.0 yes
## 1535 AB556 c16k50 Small C 237 31 2.0 yes
## 1536 AB557 c16k51 Small C 220 86 2.0 no
## 1537 AB558 c16k52 Small C 220 18 2.0 yes
## 1539 AB560 c16k54 Small C 243 75 2.0 yes
## 1549 AB591 c16k64 Small C 460 83 9.0 no
## 1552 AB608 c16k67 Small C 480 89 10.0 yes
## 1559 AB669 c16k74 Small D 490 42 10.5 no
## 1559.1 AB669 c16k74 Small D 490 42 10.5 no
## 1562 AB703 c16k77 Small D 470 68 10.0 yes
## 1565 AB753 c16k80 Small D 496 30 10.5 yes
## 1572 AB833 c16k87 Small D 470 36 10.0 yes
## 1573 AB852 c16k88 Small E 480 89 10.0 yes
## 1574 AB881 c16k89 Small E 470 32 10.0 no
## 1581 AB076 c16k96 Medium A 500 87 11.0 yes
## 1594 AB1006 c17k10 Medium A 760 124 29.0 no
## 1596 AB1009 c17k12 Medium A 600 44 17.0 yes
## 1599 AB1012 c17k15 Medium A 680 124 23.0 no
## 1603 AB1016 c17k19 Medium A 935 119 42.0 no
## 1608 AB1020 c17k24 Medium A 730 126 27.0 no
## 1612 AB1024 c17k28 Medium A 925 107 42.0 no
## 1616 AB1030 c17k32 Medium A 710 92 26.0 yes
## 1620 AB1034 c17k36 Medium A 790 54 31.0 yes
## 1623 AB1037 c17k39 Medium A 986 96 46.0 yes
## 1625 AB1043 c17k41 Medium A 990 129 47.0 yes
## 1633 AB1051 c17k49 Medium A 675 48 23.0 no
## 1639 AB1057 c17k55 Medium A 610 89 18.0 no
## 1639.1 AB1057 c17k55 Medium A 610 89 18.0 no
## 1645 AB1063 c17k61 Medium A 680 96 23.0 yes
## 1645.1 AB1063 c17k61 Medium A 680 96 23.0 yes
## 1648 AB1066 c17k64 Medium A 850 102 36.0 yes
## 1649 AB1067 c17k65 Medium A 810 97 33.0 yes
## 1651 AB1069 c17k67 Medium A 937 127 43.0 yes
## 1652 AB107 c17k68 Medium A 560 72 15.0 yes
## 1657 AB1075 c17k73 Medium A 563 64 15.0 no
## 1657.1 AB1075 c17k73 Medium A 563 64 15.0 no
## 1661 AB1079 c17k77 Medium A 557 32 14.0 yes
## 1661.1 AB1079 c17k77 Medium A 557 32 14.0 yes
## 1664 AB1081 c17k80 Medium A 814 106 33.0 no
## 1664.1 AB1081 c17k80 Medium A 814 106 33.0 no
## 1667 AB1085 c17k83 Medium A 585 59 16.0 no
## 1669 AB1087 c17k85 Medium A 568 44 15.0 yes
## 1670 AB1088 c17k86 Medium A 710 76 26.0 no
## 1673 AB1090 c17k89 Medium A 760 92 29.0 yes
## 1673.1 AB1090 c17k89 Medium A 760 92 29.0 yes
## 1677 AB1097 c17k93 Medium A 599 36 17.0 yes
## 1678 AB1099 c17k94 Medium A 580 97 16.0 yes
## 1679 AB110 c17k95 Medium A 520 69 12.0 no
## 1680 AB1100 c17k96 Medium A 560 52 15.0 no
## 1689 AB111 c18k6 Medium A 990 141 47.0 no
## 1692 AB1112 c18k9 Medium A 576 105 15.0 yes
## 1693 AB1113 c18k10 Medium A 530 61 13.0 yes
## 1694 AB1114 c18k11 Medium A 635 116 20.0 yes
## 1697 AB112 c18k14 Medium A 780 81 30.0 no
## 1705 AB1133 c18k22 Medium A 570 65 15.0 yes
## 1709 AB1138 c18k26 Medium A 637 104 20.0 no
## 1709.1 AB1138 c18k26 Medium A 637 104 20.0 no
## 1713 AB1141 c18k30 Medium A 510 44 12.0 no
## 1714 AB1142 c18k31 Medium A 645 89 21.0 no
## 1718 AB1146 c18k35 Medium A 760 84 29.0 no
## 1721 AB115 c18k38 Medium A 656 106 22.0 no
## 1721.1 AB115 c18k38 Medium A 656 106 22.0 no
## 1725 AB1155 c18k42 Medium A 610 57 18.0 no
## 1730 AB1160 c18k47 Medium A 850 113 36.0 no
## 1734 AB1164 c18k51 Medium A 753 56 29.0 yes
## 1735 AB1165 c18k52 Medium A 760 68 29.0 no
## 1737 AB1167 c18k54 Medium A 850 125 36.0 yes
## 1740 AB1170 c18k57 Medium A 720 92 26.0 yes
## 1740.1 AB1170 c18k57 Medium A 720 92 26.0 yes
## 1746 AB1177 c18k63 Medium A 859 118 37.0 yes
## 1752 AB1183 c18k69 Medium A 656 62 22.0 no
## 1752.1 AB1183 c18k69 Medium A 656 62 22.0 no
## 1763 AB1196 c18k80 Medium B 623 95 19.0 no
## 1769 AB1202 c18k86 Medium B 870 74 37.0 yes
## 1780 AB1218 c18k97 Medium B 580 102 16.0 no
## 1793 AB1239 c19k11 Medium B 550 95 14.0 no
## 1793.1 AB1239 c19k11 Medium B 550 95 14.0 no
## 1796 AB1243 c19k14 Medium B 540 74 13.0 yes
## 1802 AB1249 c19k20 Medium B 530 61 13.0 no
## 1806 AB1255 c19k24 Medium B 630 79 20.0 yes
## 1807 AB1256 c19k25 Medium B 618 70 19.0 yes
## 1809 AB1258 c19k27 Medium B 545 42 14.0 no
## 1809.1 AB1258 c19k27 Medium B 545 42 14.0 no
## 1811 AB1260 c19k29 Medium B 904 137 41.0 no
## 1812 AB1261 c19k30 Medium B 887 108 39.0 yes
## 1812.1 AB1261 c19k30 Medium B 887 108 39.0 yes
## 1813 AB1262 c19k31 Medium B 510 60 12.0 yes
## 1816 AB1265 c19k34 Medium B 589 107 16.0 yes
## 1817 AB1266 c19k35 Medium B 671 111 23.0 no
## 1823 AB1272 c19k41 Medium B 714 52 26.0 yes
## 1826 AB1275 c19k44 Medium B 640 57 21.0 no
## 1827 AB1276 c19k45 Medium B 551 87 14.0 yes
## 1828 AB1277 c19k46 Medium B 821 71 34.0 yes
## 1833 AB1282 c19k51 Medium B 662 94 22.0 yes
## 1835 AB1285 c19k53 Medium B 669 79 22.0 yes
## 1836 AB1286 c19k54 Medium B 890 60 39.0 yes
## 1841 AB1291 c19k59 Medium B 600 60 17.0 yes
## 1841.1 AB1291 c19k59 Medium B 600 60 17.0 yes
## 1845 AB1296 c19k63 Medium B 674 47 23.0 yes
## 1847 AB1298 c19k65 Medium B 619 46 19.0 yes
## 1852 AB1303 c19k70 Medium B 503 91 12.0 yes
## 1856 AB1309 c19k74 Medium B 576 97 15.0 yes
## 1860 AB1313 c19k78 Medium B 697 65 24.0 yes
## 1861 AB1314 c19k79 Medium B 980 75 46.0 yes
## 1863 AB1316 c19k81 Medium B 649 117 21.0 no
## 1866 AB1320 c19k84 Medium B 721 65 27.0 yes
## 1868 AB1322 c19k86 Medium B 667 98 22.0 yes
## 1878 AB1905 c19k96 Medium C 500 59 12.0 no
## 1879 AB1908 c19k97 Medium C 600 76 17.0 no
## 1889 AB2014 c20k8 Medium C 530 25 13.0 yes
## 1894 AB2031 c20k13 Medium C 540 74 13.0 no
## 1897 AB2048 c20k16 Medium C 530 41 13.0 yes
## 1900 AB2057 c20k19 Medium C 500 47 12.0 no
## 1900.1 AB2057 c20k19 Medium C 500 47 12.0 no
## 1907 AB2091 c20k26 Medium C 554 88 14.0 yes
## 1915 AB2138 c20k34 Medium C 520 85 12.0 yes
## 1916 AB2140 c20k35 Medium C 520 92 12.0 yes
## 1917 AB2154 c20k36 Medium C 504 87 12.0 yes
## 1917.1 AB2154 c20k36 Medium C 504 87 12.0 yes
## 1919 AB2177 c20k38 Medium C 500 63 12.0 no
## 1926 AB2206 c20k45 Medium C 520 81 12.0 yes
## 1927 AB2214 c20k46 Medium C 519 28 12.0 yes
## 1927.1 AB2214 c20k46 Medium C 519 28 12.0 yes
## 1927.2 AB2214 c20k46 Medium C 519 28 12.0 yes
## 1928 AB2222 c20k47 Medium C 590 91 16.0 no
## 1932 AB2251 c20k51 Medium C 505 51 12.0 yes
## 1939 AB2305 c20k58 Medium C 500 87 12.0 no
## 1943 AB2334 c20k62 Medium C 550 47 14.0 yes
## 1954 AB572 c20k73 Medium C 508 87 12.0 yes
## 1956 AB574 c20k75 Medium C 550 87 14.0 yes
## 1963 AB581 c20k82 Medium C 540 98 13.0 yes
## 1964 AB582 c20k83 Medium C 570 53 15.0 yes
## 1968 AB586 c20k87 Medium C 620 99 19.0 no
## 1969 AB587 c20k88 Medium C 660 118 22.0 no
## 1971 AB589 c20k90 Medium C 580 70 16.0 yes
## 1974 AB594 c20k93 Medium C 808 63 32.0 no
## 1977 AB597 c20k96 Medium C 570 73 15.0 yes
## 1977.1 AB597 c20k96 Medium C 570 73 15.0 yes
## 1984 AB606 c22k4 Medium C 500 39 12.0 yes
## 1985 AB607 c22k5 Medium C 580 82 16.0 yes
## 1988 AB611 c22k8 Medium C 640 69 21.0 no
## 1993 AB616 c22k13 Medium D 530 81 13.0 yes
## 1996 AB619 c22k16 Medium D 510 60 12.0 yes
## 1998 AB621 c22k18 Medium D 590 55 16.0 yes
## 1998.1 AB621 c22k18 Medium D 590 55 16.0 yes
## 2001 AB625 c22k21 Medium D 630 80 20.0 yes
## 2006 AB630 c22k26 Medium D 640 53 21.0 no
## 2006.1 AB630 c22k26 Medium D 640 53 21.0 no
## 2009 AB633 c22k29 Medium D 690 77 24.0 yes
## 2014 AB643 c22k34 Medium D 540 106 13.0 no
## 2018 AB647 c22k38 Medium D 710 83 25.0 yes
## 2024 AB654 c22k44 Medium D 597 108 17.0 no
## 2025 AB655 c22k45 Medium D 536 70 13.0 yes
## 2026 AB656 c22k46 Medium D 640 113 21.0 no
## 2030 AB660 c22k50 Medium D 930 127 42.0 yes
## 2042 AB675 c22k62 Medium D 600 92 17.0 yes
## 2043 AB676 c22k63 Medium D 520 101 12.0 yes
## 2047 AB682 c22k67 Medium D 650 81 22.0 yes
## 2048 AB683 c22k68 Medium D 656 90 22.0 no
## 2051 AB686 c22k71 Medium D 620 67 19.0 no
## 2055 AB690 c22k75 Medium D 750 95 28.0 yes
## 2058 AB693 c22k78 Medium D 640 93 21.0 no
## 2058.1 AB693 c22k78 Medium D 640 93 21.0 no
## 2065 AB700 c22k85 Medium D 593 95 17.0 yes
## 2067 AB702 c22k87 Medium D 520 85 12.0 no
## 2067.1 AB702 c22k87 Medium D 520 85 12.0 no
## 2069 AB705 c22k89 Medium D 540 86 13.0 no
## 2079 AB715 c22k99 Medium D 710 87 25.0 no
## 2080 AB716 c23k1 Medium D 600 104 17.0 no
## 2082 AB718 c23k3 Medium D 650 89 21.0 yes
## 2086 AB722 c23k7 Medium D 820 98 33.0 yes
## 2092 AB732 c23k13 Medium D 570 73 15.0 no
## 2098 AB738 c23k19 Medium D 760 124 29.0 yes
## 2104 AB752 c23k25 Medium D 508 55 12.0 yes
## 2108 AB758 c23k29 Medium D 650 65 21.0 yes
## 2110 AB760 c23k31 Medium D 810 130 33.0 no
## 2115 AB765 c23k36 Medium D 920 130 42.0 yes
## 2116 AB769 c23k37 Medium D 590 87 17.0 no
## 2120 AB773 c23k41 Medium D 500 51 12.0 yes
## 2121 AB774 c23k42 Medium D 550 51 14.0 no
## 2123 AB776 c23k44 Medium D 569 60 15.0 yes
## 2123.1 AB776 c23k44 Medium D 569 60 15.0 yes
## 2124 AB777 c23k45 Medium D 580 54 16.0 yes
## 2127 AB780 c23k48 Medium D 590 39 17.0 yes
## 2133 AB786 c23k54 Medium D 780 121 30.0 yes
## 2135 AB788 c23k56 Medium D 810 96 32.0 no
## 2142 AB795 c23k63 Medium D 628 91 19.0 yes
## 2142.1 AB795 c23k63 Medium D 628 91 19.0 yes
## 2144 AB797 c23k65 Medium D 810 117 33.0 no
## 2145 AB798 c23k66 Medium D 880 120 38.0 no
## 2152 AB808 c23k73 Medium D 986 151 46.0 yes
## 2154 AB810 c23k75 Medium D 520 105 12.0 yes
## 2157 AB814 c23k78 Medium D 551 79 14.0 yes
## 2159 AB816 c23k80 Medium D 620 54 19.0 yes
## 2169 AB826 c23k90 Medium D 750 75 28.0 no
## 2170 AB828 c23k91 Medium D 545 102 14.0 yes
## 2174 AB832 c23k95 Medium D 711 68 26.0 no
## 2174.1 AB832 c23k95 Medium D 711 68 26.0 no
## 2181 AB840 c24k3 Medium E 694 73 24.0 yes
## 2184 AB843 c24k6 Medium E 710 91 25.0 no
## 2187 AB846 c24k9 Medium E 750 51 28.0 no
## 2189 AB848 c24k11 Medium E 892 92 40.0 yes
## 2191 AB850 c24k13 Medium E 564 32 15.0 yes
## 2193 AB853 c24k15 Medium E 582 98 16.0 no
## 2195 AB855 c24k17 Medium E 830 64 35.0 yes
## 2199 AB859 c24k21 Medium E 810 105 33.0 yes
## 2204 AB867 c24k26 Medium E 546 34 14.0 yes
## 2217 AB882 c24k39 Medium E 510 99 12.0 no
## 2221 AB886 c24k43 Medium E 605 69 18.0 yes
## 2224 AB889 c24k46 Medium E 855 90 36.0 yes
## 2230 AB898 c24k52 Medium E 670 115 23.0 yes
## 2231 AB899 c24k53 Medium E 750 75 28.0 no
## 2231.1 AB899 c24k53 Medium E 750 75 28.0 no
## 2233 AB901 c24k55 Medium E 868 62 37.0 yes
## 2234 AB902 c24k56 Medium E 630 91 20.0 yes
## 2247 AB921 c24k69 Medium E 531 93 13.0 no
## 2252 AB926 c24k74 Medium E 640 100 20.0 yes
## 2255 AB929 c24k77 Medium E 758 48 29.0 yes
## 2255.1 AB929 c24k77 Medium E 758 48 29.0 yes
## 2263 AB938 c24k85 Medium E 520 52 12.0 no
## 2267 AB942 c24k89 Medium E 610 86 18.0 no
## 2272 AB947 c24k94 Medium E 700 118 24.0 no
## 2279 AB955 c25k2 Medium E 944 64 43.0 no
## 2282 AB961 c25k5 Medium E 590 79 16.0 yes
## 2284 AB964 c25k7 Medium E 540 66 13.0 yes
## 2285 AB965 c25k8 Medium E 601 65 18.0 no
## 2296 AB976 c25k19 Medium E 910 109 41.0 yes
## 2304 AB990 c25k27 Medium E 571 77 15.0 yes
## 2305 AB991 c25k28 Medium E 510 104 12.0 no
## 2305.1 AB991 c25k28 Medium E 510 104 12.0 no
## 2306 AB992 c25k29 Medium E 610 93 18.0 yes
## 2313 AB999 c25k36 Medium E 648 65 21.0 no
## 2316 AB1008 c25k39 Big A 1390 171 77.0 yes
## 2317 AB1026 c25k40 Big A 1044 78 52.0 yes
## 2319 AB1029 c25k42 Big A 1110 97 58.0 yes
## 2322 AB1040 c25k45 Big A 1130 82 61.0 yes
## 2322.1 AB1040 c25k45 Big A 1130 82 61.0 yes
## 2330 AB1108 c25k53 Big A 1094 83 57.0 yes
## 2335 AB1125 c25k58 Big A 1450 133 94.0 yes
## 2336 AB1126 c25k59 Big A 1614 159 138.0 yes
## 2337 AB1130 c25k60 Big A 1080 140 54.0 yes
## 2340 AB1153 c25k63 Big A 1121 146 59.0 yes
## 2341 AB1172 c25k64 Big A 1440 133 84.0 yes
## 2346 AB1209 c25k69 Big B 1060 90 53.0 no
## 2347 AB1211 c25k70 Big B 1050 142 52.0 yes
## 2348 AB1212 c25k71 Big B 1616 158 142.0 yes
## 2348.1 AB1212 c25k71 Big B 1616 158 142.0 yes
## 2350 AB1220 c25k73 Big B 1480 193 104.0 no
## 2350.1 AB1220 c25k73 Big B 1480 193 104.0 no
## 2350.2 AB1220 c25k73 Big B 1480 193 104.0 no
## 2351 AB1231 c25k74 Big B 1160 121 62.0 no
## 2359 AB637 c25k82 Big D 1405 111 83.0 yes
## 2361 AB725 c25k84 Big D 1005 83 49.0 yes
## 2371 AB766 c25k94 Big D 1110 93 58.0 yes
## 2372 AB767 c25k95 Big D 1300 176 68.0 yes
## 2372.1 AB767 c25k95 Big D 1300 176 68.0 yes
## 2373 AB799 c25k96 Big D 1094 131 56.0 no
## 2378 AB891 c26k2 Big E 1040 118 52.0 no
## 2378.1 AB891 c26k2 Big E 1040 118 52.0 no
## 2379 AB892 c26k3 Big E 1084 92 54.0 yes
## 2382 AB905 c26k6 Big E 1405 110 83.0 no
## 2387 AB932 c26k11 Big E 1360 104 72.0 yes
## 2390 AB957 c26k14 Big E 1220 163 63.0 no
## 2392 AB984 c26k16 Big E 1020 89 50.0 yesa.estimar <- data.frame(muestra$Income, muestra$Employees, muestra$Taxes);E.WR(N,n,a.estimar)## N muestra.Income muestra.Employees muestra.Taxes
## Estimation 2396 1.067506e+06 1.547465e+05 30403.761836
## Standard Error 0 2.492457e+04 2.939407e+03 1601.824661
## CVE 0 2.334842e+00 1.899498e+00 5.268508
## DEFF NaN 1.410241e+00 1.410241e+00 1.4102419.3.3 El efecto de diseño
\[ \begin{align} {Deff}_{MAS_R}&=\frac{{V}_{MAS_R}\left(\widehat{T}_{y,\cdot}\right)}{{V}_{MAS}\left(\widehat{T}_{y,\pi}\right)}\\ &=\frac{{V}_{MAS_R}\left(\widehat{t}_{y,\pi}\right)}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\frac{N\left(N-1\right)}{m}S_{yU}^{2}}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\frac{N^2}{m}\left(1-\frac{1}{N}\right)S_{yU}^{2}}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)S_{yU}^{2}}\\ &=\frac{\frac{N^2}{m}\left(1-\frac{1}{N}\right)}{\frac{N^2}{n}\left(1-\frac{n}{N}\right)}\\ &=\frac{\frac{1}{m}\left(1-\frac{1}{N}\right)}{\frac{1}{n}\left(1-\frac{n}{N}\right)}\\ &=\left(1-\frac{1}{N}\right)\frac{1}{\frac{m}{n}\left(1-\frac{n}{N}\right)}\\ &=\left(1-\frac{1}{N}\right)\frac{1}{\left(1-\frac{n}{N}\right)}\\ \end{align} \]
Deff_MAS_R <- (1-1/N)*(1/(1-n/N)); Deff_MAS_R## [1] 1.409653n*Deff_MAS_R## [1] 982.5289.4 Diseño de muestreo sistemático
\[N=na+c\]
\[0{\leq}c{<}a\]
\[c=N-\left\|\frac{N}{a}\right\|a\]
\[s=\left\{r,r+1a,r+2a,\ldots,r+(n-1)a\right\}\]
| Grupo | \(s_1\) | \(\cdots\) | \(s_r\) | \(\cdots\) | \(s_a\) |
|---|---|---|---|---|---|
| \(n=1\) | \(1\) | \(\cdots\) | \(r\) | \(\cdots\) | \(a\) |
| \(n=2\) | \(1+a\) | \(\cdots\) | \(r+a\) | \(\cdots\) | \(2a\) |
| \(n=3\) | \(1+2a\) | \(\cdots\) | \(r+2a\) | \(\cdots\) | \(3a\) |
| \(\vdots\) | \(\vdots\) | \(\ddots\) | \(\vdots\) | \(\ddots\) | \(\vdots\) |
| \(n=\left\|\frac{N}{a}\right\|\) | \(1+(n-1)a\) | \(\cdots\) | \(r+(n-1)a\) | \(\cdots\) | \(na\) |
| \(n=\left\|\frac{N}{a}\right\|+1\) | \(1+na\) | \(\cdots\) | \(\square\) | \(\cdots\) | \(\square\) |
\[ \begin{align} U&={\bigcup}_{r=1}^{a}s_r \end{align} \]
\[Q_r=\left\{s_1,s_2,\ldots,s_r,\ldots,s_a\right\}\]
\[\#\left(Q_r\right)=a\]
\[ \begin{align} \mathcal{p}(s)&= \begin{cases} \frac{1}{a}&\text{ si }s{\in}Q_r\\ 0&\text{ en otro caso} \end{cases} \end{align} \]
9.4.1 Algoritmo de selección
- \({0}{\leq}p(r)={\frac{1}{a}}{\leq}{1}\)
\(\forall_{k{\in}\left\{1,2,\ldots,a\right\}}r{\sim}U{\left[1,a\right]}\)
\(r{\in}s\)
- \(s_{r} = \left\{k: k = r + (j-1)a; j=1,2,\ldots,n(s)\right\}\)
\[U=\{Santiago, Nestor, Nayibe, Raul, Jhon\}\]
| Grupo | \(s_1\) | \(s_2\) |
|---|---|---|
| \(n=1\) | \(Santiago\) | \(Raul\) |
| \(n=2\) | \(Nestor\) | \(Jhon\) |
| \(n=\left\|\frac{5}{2}\right\|+1\) | \(Nayibe\) | \(\square\) |
\(s_1=\{Santiago, Nestor, Nayibe\}\)
\(s_2=\{Raul, Jhon\}\)
\[ \begin{align} \pi_k&={\sum}_{s{\in}Q}I_k\left({s}\right)\mathcal{p}\left({s}\right)\\ &=\frac{\binom{1}{1}\binom{a-1}{0}}{\binom{a}{1}}\\ &=\frac{1}{a} \end{align} \]
\[ \begin{align} \pi_{kl}&=\mathcal{P}(k{\in}s_r\text{ & }l{\in}s_r)\\ &=\mathcal{P}(I_k(S_r)=1{\mid}I_l(S_r)=1)\mathcal{P}(I_l(S_r)=1)\\ &=1\frac{1}{a}\\ &=\frac{1}{a} \end{align} \]
9.4.2 Estimador de Horvitz - Thompson
\[ \begin{align} \widehat{t}_{y,\pi}&={\sum}_{s}\frac{y_k}{\pi_k}\\ &=\frac{1}{\pi_k}{\sum}_{s}y_k\\ &=\frac{1}{\frac{1}{a}}{\sum}_{s}y_k\\ &=\frac{a}{1}{\sum}_{s}y_k \end{align} \]
\[ \begin{align} V[\widehat{t}_{y,\pi}]&={{\sum}{\sum}}_{U}{\Delta}_{kl}\frac{y_k}{\pi_k}\frac{y_l}{\pi_l} \end{align} \]
\[ \begin{align} {\Delta}_{kl}&= \begin{cases} {\pi}_{kl}-{\pi}_{k}{\pi}_{l}&\text{ para }k{\neq}l\\ {\pi}_{k}\left(1-{\pi}_{k}\right)&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} 0&\text{ para }k{\neq}l\\ \frac{1}{a}-\frac{1}{a}\frac{1}{a}&\text{ para }k{=}l \end{cases}\\ &= \begin{cases} 0&\text{ para }k{\neq}l\\ \frac{1}{a}\left(1-\frac{1}{a}\right)&\text{ para }k{=}l \end{cases} \end{align} \]
\[ \begin{align} V_{SIS}[\widehat{t}_{y,\pi}]&=V_{SIS}\left[\frac{a}{1}{\sum}_{s}y_k\right]\\ &=\frac{a^2}{1^2}V_{SIS}\left[{\sum}_{U}I_k(s)y_k\right]\\ &=\frac{a^2}{1^2}V_{SIS}\left[{\sum\sum}_{U}I_k(s)y_k\right]\\ &=\frac{a^2}{1^2}{\sum\sum}_{U}C_{SIS}\left[I_k(s)y_k,I_l(s)y_l\right]\\ &=\frac{a^2}{1^2}\left\{{\sum}_{k=l}V_{SIS}\left[I_k(s)y_k\right]+{\sum\sum}_{k{\neq}l}C_{SIS}\left[I_k(s)y_k,I_l(s)y_l\right]\right\}\\ &=\frac{a^2}{1^2}\left\{{\sum}_{k=l}V_{SIS}\left[I_k(s)\right]y_k^2+{\sum\sum}_{k{\neq}l}C_{SIS}\left[I_k(s),I_l(s)\right]y_ky_l\right\}\\ &=\frac{a^2}{1^2}\left\{{\sum}_{k=l}\frac{1}{a^2}\left(N-n\right)y_k^2-{\sum\sum}_{k{\neq}l}\frac{1}{a^2}\left[\frac{a-1}{a-1}\right]y_ky_l\right\}\\ &=\frac{a^2}{1^2}\left\{\frac{1}{a^2}\left(a-1\right){\sum}_{k=l}y_k^2-\frac{1}{a^2}\left[\frac{a-1}{a-1}\right]{\sum\sum}_{k{\neq}l}y_ky_l\right\}\\ &=\frac{a^2}{1^2}\frac{1}{a^2}\left(a-1\right)\left\{{\sum}_{k=l}y_k^2-\frac{1}{N-1}{\sum\sum}_{k{\neq}l}y_ky_l\right\}\\ &=\frac{1}{1}\left(a-1\right)\left\{{\sum}_{U}y_k^2-\frac{1}{a-1}\left[\left({{\sum}_{U}y_k}\right)^2-{\sum}_{U}y_k^2\right]\right\}\\ &=\frac{1}{1}\left(a-1\right)\frac{1}{a-1}\left\{\left(a-1\right){\sum}_{U}y_k^2-\left[\left({{\sum}_{U}y_k}\right)^2-{\sum}_{U}y_k^2\right]\right\}\\ &=\frac{1}{1}\left(a-1\right)\frac{1}{a-1}\left\{a{\sum}_{U}y_k^2-{\sum}_{k=l}y_k^2-\left({{\sum}_{U}y_k}\right)^2+{\sum}_{U}y_k^2\right\}\\ &=\frac{1}{1}\left(a-1\right)\frac{1}{a-1}\left\{N{\sum}_{U}y_k^2-\left({{\sum}_{U}y_k}\right)^2\right\}\\ &=\frac{a}{1}\left(a-1\right)\frac{1}{a-1}\left\{{\sum}_{U}y_k^2-\frac{1}{a}\left({{\sum}_{U}y_k}\right)^2\right\}\\ &=\frac{a^2}{1}\left(1-\frac{1}{a}\right)\frac{1}{a-1}\left\{{\sum}_{U}y_k^2-{N}\left[\frac{{\sum}_{U}y_k}{a}\right]^2\right\}\\ &=\frac{a^2}{1}\left(1-\frac{1}{a}\right)S_{yU}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{SIS}[\widehat{t}_{y,\pi}]&=\frac{a^2}{1}\left(1-\frac{1}{a}\right)S_{ys}^{2} \end{align} \]
\[ \begin{align} E\left[S_{ys}^{2}\right]&=E\left\{\frac{1}{a-1}\left[{\sum}_{s}y_k^2-{a}\bar{y}_{s}^2\right]\right\}\\ &=\frac{1}{a-1}E\left[{\sum}_{s}y_k^2-{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left\{E\left[{\sum}_{s}y_k^2\right]-{a}E\left[\bar{y}_{s}^2\right]\right\}\\ &=\frac{1}{a-1}\left\{E\left[{\sum}_{U}I_k(s)y_k^2\right]-{a}E\left[\frac{\widehat{t}_{y,\pi}^2}{a^2}\right]\right\}\\ &=\frac{1}{a-1}\left\{{\sum}_{U}E\left[I_k(s)\right]y_k^2-\frac{a}{a^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{1}{a-1}\left\{{\sum}_{U}\pi_ky_k^2-\frac{a}{a^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{1}{a-1}\left\{\frac{a}{a}{\sum}_{U}y_k^2-\frac{a}{a^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a^2}\left\{V_{MAS}\left[\widehat{t}_{y,\pi}^2\right]-{t}_{y,\pi}^{2}\right\}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a^2}\frac{a^2}{a}\left[1-\frac{a}{a}\right]S_{yU}^{2}-\frac{1}{a^2}{t}_{y,\pi}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a}\left[1-\frac{a}{a}\right]S_{yU}^{2}-\frac{1}{a^2}{t}_{y,\pi}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a^2}{t}_{y,\pi}^{2}-\frac{1}{a}\left[\frac{a}{a}-\frac{a}{a}\right]S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}\left[{\sum}_{U}y_k^2-\frac{1}{a}{t}_{y,\pi}^{2}\right]-\frac{1}{a}\left[\frac{a-a}{a}\right]S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{a-1}{a}\frac{1}{a-1}\left[{\sum}_{U}y_k^2-{a}\frac{{t}_{y,\pi}^{2}}{a^{2}}\right]-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{a-1}{a}\frac{1}{a-1}\left[{\sum}_{U}y_k^2-{a}\bar{y}_{U}^2\right]-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{a-1}{a}S_{yU}^{2}-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{aa-a}{aa}S_{yU}^{2}-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\frac{aa-a-(a-a)}{aa}S_{yU}^{2}\\ &=\frac{a}{a-1}\frac{aa-a-a+a}{aa}S_{yU}^{2}\\ &=\frac{a}{a-1}\frac{aa-a}{aa}S_{yU}^{2}\\ &=\frac{a}{a-1}\frac{a-1}{a}S_{yU}^{2}\\ &=S_{yU}^{2} \end{align} \]
\[ \begin{align} S_{ys}^{2}&=\frac{1}{a-1}\left\{{\sum}_{s}y_k^2-{a}\left[\frac{{\sum}_{s}y_k}{a}\right]^2\right\}\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2{a}\bar{y}_{s}^2+{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{a}\bar{y}_{s}+{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{a}\frac{{\sum}_{s}y_k}{a}+{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{\sum}_{s}y_k+{\sum}_{s}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}{\sum}_{s}\left(y_k^2-2y_k\bar{y}_{s}+\bar{y}_{s}^2\right)\\ &=\frac{1}{a-1}{\sum}_{s}\left(y_k-\bar{y}_{s}\right)^2\\ \end{align} \]
\[ \begin{align} \widehat{t}_{y,\pi}(s=U)&={\sum}_{s=U}\frac{y_k}{\pi_k}\\ &=\frac{a}{a}{\sum}_{s=U}y_k\\ &={t}_{y} \end{align} \]
\[ \begin{align} \widehat{t}_{y,alt}&=a\tilde{y}_{s}\\ &=a\frac{{\sum}_{s}\frac{y_k}{{\pi}_{k}}}{{\sum}_{s}\frac{1}{{\pi}_{k}}}\\ &=a\frac{\widehat{t}_{y\pi}}{\widehat{a}}\\ &=a\frac{\widehat{t}_{y\pi}}{a{(s)}}\\ &=a\bar{y}_{s}\\ \end{align} \]
9.4.2.1 Estimador del promedio
\[ \begin{align} \widehat{\bar{y}}_{\pi}&=\frac{\widehat{t}_{y,\pi}}{a}\\ &=\frac{{\sum}_{s}\frac{y_k}{{\pi}_{k}}}{a}\\ &=\frac{\frac{a}{a}{\sum}_{s}{y_k}}{\frac{a}{1}}\\ &=\frac{{\sum}_{s}{y_k}}{a}\\ &=\bar{y}_{s}\\ \end{align} \]
\[ \begin{align} {V}_{SIS}[\widehat{\bar{y}}_{\pi}]&=\frac{a^2}{a}\left(1-\frac{a}{a}\right)\frac{1}{a^2}S_{yU}^{2}\\ &=\frac{1}{a}\left(1-\frac{a}{a}\right)S_{yU}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{SIS}[\widehat{\bar{y}}_{\pi}]&=\frac{a^2}{a}\left(1-\frac{a}{a}\right)\frac{1}{a^2}S_{ys}^{2}\\ &=\frac{1}{a}\left(1-\frac{a}{a}\right)S_{ys}^{2} \end{align} \]
\[ \begin{align} E\left[S_{ys}^{2}\right]&=E\left\{\frac{1}{a-1}\left[{\sum}_{s}y_k^2-{a}\bar{y}_{s}^2\right]\right\}\\ &=\frac{1}{a-1}E\left[{\sum}_{s}y_k^2-{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left\{E\left[{\sum}_{s}y_k^2\right]-{a}E\left[\bar{y}_{s}^2\right]\right\}\\ &=\frac{1}{a-1}\left\{E\left[{\sum}_{U}I_k(s)y_k^2\right]-{a}E\left[\frac{\widehat{t}_{y,\pi}^2}{a^2}\right]\right\}\\ &=\frac{1}{a-1}\left\{{\sum}_{U}E\left[I_k(s)\right]y_k^2-\frac{a}{a^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{1}{a-1}\left\{{\sum}_{U}\pi_ky_k^2-\frac{a}{a^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{1}{a-1}\left\{\frac{a}{a}{\sum}_{U}y_k^2-\frac{a}{a^2}E\left[\widehat{t}_{y,\pi}^2\right]\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a^2}\left\{V_{MAS}\left[\widehat{t}_{y,\pi}^2\right]-{t}_{y,\pi}^{2}\right\}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a^2}\frac{a^2}{a}\left[1-\frac{a}{a}\right]S_{yU}^{2}-\frac{1}{a^2}{t}_{y,\pi}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a}\left[1-\frac{a}{a}\right]S_{yU}^{2}-\frac{1}{a^2}{t}_{y,\pi}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}{\sum}_{U}y_k^2-\frac{1}{a^2}{t}_{y,\pi}^{2}-\frac{1}{a}\left[\frac{a}{a}-\frac{a}{a}\right]S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{1}{a}\left[{\sum}_{U}y_k^2-\frac{1}{a}{t}_{y,\pi}^{2}\right]-\frac{1}{a}\left[\frac{a-a}{a}\right]S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{a-1}{a}\frac{1}{a-1}\left[{\sum}_{U}y_k^2-{a}\frac{{t}_{y,\pi}^{2}}{a^{2}}\right]-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{a-1}{a}\frac{1}{a-1}\left[{\sum}_{U}y_k^2-{a}\bar{y}_{U}^2\right]-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{a-1}{a}S_{yU}^{2}-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\left\{\frac{aa-a}{aa}S_{yU}^{2}-\frac{a-a}{aa}S_{yU}^{2}\right\}\\ &=\frac{a}{a-1}\frac{aa-a-(a-a)}{aa}S_{yU}^{2}\\ &=\frac{a}{a-1}\frac{aa-a-a+a}{aa}S_{yU}^{2}\\ &=\frac{a}{a-1}\frac{aa-a}{aa}S_{yU}^{2}\\ &=\frac{a}{a-1}\frac{a-1}{a}S_{yU}^{2}\\ &=S_{yU}^{2} \end{align} \]
\[ \begin{align} S_{ys}^{2}&=\frac{1}{a-1}\left\{{\sum}_{s}y_k^2-{a}\left[\frac{{\sum}_{s}y_k}{a}\right]^2\right\}\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2{a}\bar{y}_{s}^2+{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{a}\bar{y}_{s}+{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{a}\frac{{\sum}_{s}y_k}{a}+{a}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}\left[{\sum}_{s}y_k^2-2\bar{y}_{s}{\sum}_{s}y_k+{\sum}_{s}\bar{y}_{s}^2\right]\\ &=\frac{1}{a-1}{\sum}_{s}\left(y_k^2-2y_k\bar{y}_{s}+\bar{y}_{s}^2\right)\\ &=\frac{1}{a-1}{\sum}_{s}\left(y_k-\bar{y}_{s}\right)^2\\ \end{align} \]
\[ \begin{align} \widehat{t}_{y,\pi}(s=U)&={\sum}_{s=U}\frac{y_k}{\pi_k}\\ &=\frac{a}{a}{\sum}_{s=U}y_k\\ &={t}_{y} \end{align} \]
\[ \begin{align} \widehat{t}_{y,alt}&=a\tilde{y}_{s}\\ &=a\frac{{\sum}_{s}\frac{y_k}{{\pi}_{k}}}{{\sum}_{s}\frac{1}{{\pi}_{k}}}\\ &=a\frac{\widehat{t}_{y\pi}}{\widehat{a}}\\ &=a\frac{\widehat{t}_{y\pi}}{n{(s)}}\\ &=a\bar{y}_{s}\\ \end{align} \]
9.4.2.2 Estimador del promedio
\[ \begin{align} \widehat{\bar{y}}_{\pi}&=\frac{\widehat{t}_{y,\pi}}{a}\\ &=\frac{{\sum}_{s}\frac{y_k}{{\pi}_{k}}}{a}\\ &=\frac{\frac{a}{a}{\sum}_{s}{y_k}}{\frac{a}{1}}\\ &=\frac{{\sum}_{s}{y_k}}{a}\\ &=\bar{y}_{s}\\ \end{align} \]
\[ \begin{align} {V}_{SIS}[\widehat{\bar{y}}_{\pi}]&=\frac{a^2}{a}\left(1-\frac{a}{a}\right)\frac{1}{a^2}S_{yU}^{2}\\ &=\frac{1}{a}\left(1-\frac{a}{a}\right)S_{yU}^{2} \end{align} \]
\[ \begin{align} \widehat{V}_{SIS}[\widehat{\bar{y}}_{\pi}]&=\frac{a^2}{a}\left(1-\frac{a}{a}\right)\frac{1}{a^2}S_{ys}^{2}\\ &=\frac{1}{a}\left(1-\frac{a}{a}\right)S_{ys}^{2} \end{align} \]