In hudie-lab/SC19036: The R Package For StatComp

Question 1

Answer 1

As here a mean of sample $X \sim \chi^2_2$ is estimated by a 95% symmetric t-interval, then we can get $\frac{\bar{X}-\mu}{\sqrt{\frac{\sum{(X_i-\bar{X})^2}}{n-1}}} \sim t_{n-1}$, so the interval estimated would be $P(\frac{\bar{X}-\mu}{\sqrt{\frac{\sum{(X_i-\bar{X})^2}}{n-1}}} > t_{n-1}(\alpha))$, which could be transformed as $P(\bar(X)>\mu+\sqrt{\frac{\sum{(X_i-\bar{X})^2}}{n-1}}t_{n-1}(\alpha))$. And we use Monte-Carlo method to get empirical confidence level.

set.seed(19036)
n <- 20
m=10000
alpha <- .05
UCL <- replicate(1000, expr = {
x <- rnorm(n, mean = 0, sd = 2)
(n-1) * var(x) / qchisq(alpha, df = n-1) })

y=rchisq(n,df=2)
ybar=mean(y)/qt(df=n-1,1-alpha)
esti_int=replicate(10000,expr={
  y=rchisq(n,df=2)
  (mean(y)-sd(y)*qt(df=n-1,alpha))
  #(mean(y)-2)/sd(y)
})
##the empirical estimate of intervals for variance in example6.4
(mean_x=mean(UCL>4))
(sd_x=sd(UCL>4))
##the empirical estimate of intervals for mean 
(mean_t=mean(esti_int>2))
(sd_t=sd(esti_int>2))

From the results, we can see that the t-interval estimate is much larger than the $\alpha$ which means that this method is much more robust.

Qestion 2

Answer 2

estimate of the quantiles of the skewness

n=300
p=c(0.025,0.05,0.95,0.975)
##true quantile of the skewness distribution
cp=qnorm(p,sd=sqrt(6/n))

m=10000
sb1=replicate(m,expr = {
  x=rnorm(n)
  1/n*sum((x-mean(x))^3)/(((n-1)/n*var(x))^(3/2))
})
esti_b1=rep(0,length(cp))
for(l in 1:length(cp)){
  esti_b1[l]=quantile(sb1,p[l])
}
##estimate of the quantile of the skewness distribution
esti_b1

knitr::kable(rbind(cp,esti_b1),col.names = p)

The table above is the comparison of the estimated quantiles with the true quantiles of $\sqrt{b_1}$.

compute the estimate of the variance in $(2.14)$

sd1=rep(0,length(cp))
for(l in 1:length(cp)){
  sd1[l]=p[l]*(1-p[l])/n/(dnorm(cp[l],sd=sqrt(6/n)))^2
}
sd1

I compute the $f(x_q)$ by plug the true quantiles into the density function of $\sqrt{b_1}$.

hudie-lab/SC19036 documentation built on Jan. 2, 2020, 8:45 p.m.