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# 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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