Técnicas de remuestreo

Realizar inferencia sobre un parámetro con la distribución no analítica.

Leave one Out

\(\theta\)

\(\theta_{-1}\) \(\theta_{-2}\) \(\vdots\) \(\theta_{-n}\)

Jacknife(K=2)

\(\theta\)

\(\theta_{-1,-2}\) \(\theta_{-1,-3}\) \(\vdots\) \(\theta_{-(n-1),-n}\)

Jacknife(K=2)

La cantidad de submuetsras que se tienen es \[ \binom{n}{k} \] Se hace uso de montecarlo para hacer IC

Método de montecarlo.

n=100
x<-runif(n,-1,1)
y<-runif(n,-1,1)
plot(x,y)

U<-x**2+y**2 <=1
mean(U)
## [1] 0.72
mean(U)*4
## [1] 2.88
n=200
x<-runif(n,-1,1)
y<-runif(n,-1,1)
plot(x,y)

U<-x**2+y**2 <=1
mean(U)
## [1] 0.775
mean(U)*4
## [1] 3.1
n=500
x<-runif(n,-1,1)
y<-runif(n,-1,1)
plot(x,y)

U<-x**2+y**2 <=1
mean(U)
## [1] 0.806
mean(U)*4
## [1] 3.224
n=1000
x<-runif(n,-1,1)
y<-runif(n,-1,1)
plot(x,y)

U<-x**2+y**2 <=1
mean(U)
## [1] 0.779
mean(U)*4
## [1] 3.116
n=3000
x<-runif(n,-1,1)
y<-runif(n,-1,1)
plot(x,y)

U<-x**2+y**2 <=1
mean(U)
## [1] 0.8203333
mean(U)*4
## [1] 3.281333
n=10000
x<-runif(n,-1,1)
y<-runif(n,-1,1)
plot(x,y)

U<-x**2+y**2 <=1
mean(U)
## [1] 0.7847
mean(U)*4
## [1] 3.1388
n=50000
x<-runif(n,-1,1)
y<-runif(n,-1,1)
plot(x,y)

U<-x**2+y**2 <=1
mean(U)
## [1] 0.78546
mean(U)*4
## [1] 3.14184
pi=NULL
for(i in seq(50,20000,by=50)){
n=i
x<-runif(n,-1,1)
y<-runif(n,-1,1)
U<-x**2+y**2 <=1
mean(U)
mean(U)*4
  pi[i/50]<- mean(U)*4
}
plot(seq(50,20000,by=50),pi, type="l")

\[\nu =\frac{\sigma_X}{|\mu_X|}= 1\]

n=1000
x<-rnorm(n,1,1)
# Leave One Out
X<-matrix(0,n,n-1)
for(i in 1:n){
  X[i,]<-x[-i]
}
cv<-rep(0,n)
for(i in 1:n){cv[i]<-sd(X[i,])/abs(mean(X[i,]))}
IC<-c(mean(cv)-1.96*sd(cv),mean(cv)+1.96*sd(cv))
IC
## [1] 0.9755032 0.9803101
plot(cv,type="l")