set.seed(123)
N <- 100
n <- 4
k <- N/n
B <- 200
alpha <- 0.05
muestra <- rexp(N,1)
xbar <- mean(muestra)
submues <- matrix(muestra,nrow = k,ncol = n)
xbarsub <- apply(submues, 1, mean) 
s2sub <- apply(submues, 1, var) 
xbarbar <- mean(xbarsub)
S2 <- mean(s2sub) 
S <- sqrt(S2)
plot(xbarsub,type="l")
points(xbarsub)
abline(h=xbarbar, col = "red")

Para los límites de control asintóticos, se generaron:

##Asintótico
LIas <- xbarbar-qnorm(1-alpha/2)*(S/sqrt(n))
LSas <- xbarbar+qnorm(1-alpha/2)*(S/sqrt(n))
LIas #Limite inferior
## [1] 0.06678477
LSas #Limite superior
## [1] 2.024653
plot(xbarsub,type="l")
points(xbarsub)
abline(h=xbarbar, col = "red")
abline(h=LIas, col = "green")
abline(h=LSas, col = "green")

Sean \(x_{1},x_{2},...,x_{n}\) variableas aleatorias tales que \(x_{i}\sim Exp(\lambda), \ \ ( \forall i \in [1,n])\), entonces \(\sum_{i=1}^{n} x_{i} \sim Gamma(n,\lambda)\)

Ahora, recordemos que para una cte positiva (\(C>0\)), se tiene que: \[X \sim Gamma(n,\lambda), \ \ entonces \ \ CX \sim Gamma(n,C\lambda)\].

Por lo tanto: \[Y = \frac{\sum_{i=1}^{n} x_{i}}{n} \sim Gamma(n,\frac{\lambda}{n})\]

\(E[Y] = n*\frac{\lambda}{n} = \lambda = 1\)

\(Var[Y] = n*\frac{\lambda^{2}}{n^{2}} =\frac{\lambda^{2}}{n} = \frac{1}{4}\)

##Exacto
mu <-1
sigma <- 1/4

LIex <- mu-qgamma(alpha,4,1/4)*(sigma/sqrt(n))
LSex <- mu+qgamma(alpha,4,1/4)*(sigma/sqrt(n))

LIex #Limite inferior 
## [1] 0.3168408
LSex #Limite superior
## [1] 1.683159
plot(xbarsub,type="l")
points(xbarsub)
abline(h=xbarbar, col = "red")
abline(h=LIex, col = "blue")
abline(h=LSex, col = "blue")

###Boot
stat.boot <- matrix(NA,nrow = B,ncol = k)
xbarsub.boot <- matrix(NA,nrow = B,ncol = k)
xbarbar.boot <- vector()
for(i in 1:B){
muestra.boot <- sample(muestra,replace = T)
submues.boot <- matrix(muestra.boot,nrow = k,ncol = n)
xbarsub.boot[i,] <- apply(submues.boot, 1, mean) 
xbarbar.boot[i] <- mean(xbarsub.boot[i,])

stat.boot[i,] <- sqrt(n)*(xbarsub.boot[i]-xbarbar)
}
hist(stat.boot)

t <- quantile(stat.boot,c(1-alpha))
LIboot <- mean(xbarbar.boot)-t[1]/sqrt(n)
LSboot <- mean(xbarbar.boot)+t[1]/sqrt(n)
LIboot #Limite inferior
##       95% 
## 0.1015447
LSboot #Limite superior
##      95% 
## 1.981437