Approximation theorems of mathmatical statics
#Approximation theorems of mathmatical statics p.64
#cramer-von Mises test
library(dplyr);library(mvnorm)
n=10000
x=rbinom(n, size=50, prob=0.3)
values=sort(unique(x))
data=data.frame(values=values,Fn=0,F=0,f=0,fn=0)
for(j in 1:length(values)){
data$Fn[j]=sum(x<data$values[j])/n
data$F[j]=ppois(data$values[j],50*0.3)
data$f[j]=dpois(data$values[j],50*0.3)
#sample density functions(bn=1/log(n))
data$fn[j]=(sum(x<data$values[j]+1/log(n))/n-sum(x<data$values[j]-1/log(n))/n)/(2/log(n))
}
C_n=n*sum(data$f*(data$Fn-data$F)^2)
#using theorem A(Finkelstein)
epsilon=0.00001
under=C_n-(1+epsilon)*(2*log(log(n)))/(pi^2)
upper=C_n+(1+epsilon)*(2*log(log(n)))/(pi^2)
#p.68 2.2.2
n=10000
mu=2
x=rnorm(n,mu,1)
b=c()
mu2=c()
k=10
for(j in 1:k){
b=c(b,sum((x-mu)^j)/n)
mu2=c(mu2,sum((x-mu)^(2*j))/n)
}
#V(bk)
(mu2-b^2)/n
cov_mat=array(0,dim=c(k,k))
for(i in 1:k){
for(j in 1:k){
cov_mat[i,j]=(sum((x-mu)^(i+j))/n-sum((x-mu)^i)*sum((x-mu)^j)/(n^2))/n
}}
(rmvnorm(100,b,cov_mat),2,mean)
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