Example: Bootstrap try to catch the skewness in the sampling distribution
Sta662 Class notes 22 Oct 2007


T-statistic is commonly used in testing and constructing of confidence intervals. 
When the sample x1, x2, ..., xn is truly iid from a Normal population, 
then it is an exact distribution:

sqrt(n) [X bar - mean(F)] / s        is (student) t with df = n-1


However, when the population is not Normal, for example exp( ) 
then the t-distribution missed the skewness of the statistic

> for(i in 1:9000) { xvec <- rexp(20)
 result[i] <- sqrt(20)*( mean(xvec)-1 )/sd(xvec)}
> hist(result[1:9000])

We see it is skewed. (to the left or to the right?, try without the numerator)
If we use a t-distribution, we are using a symmetric distribution (t) to approximate 
a skewed distribution. We miss the skewness. 
Bootstrap do not pre-fix a symmetric distribution and let the sample tell the story, 
and here it captures the skewness in the t-statistic:

> xvec <- rexp(20)
> hist(xvec)
>
> for(i in 1:9000) { Bxvec <- sample(xvec, size=20, replace=TRUE)
 result[i] <- sqrt(20)*( mean(Bxvec)- mean(xvec) )/sd(Bxvec)}
> 
> hist(result)




Some comments: the skewness converges to zero as sample size n go to infinite. (central limit theorem). 
The error caused by skewness is of the order 1/sqrt(n). If we use rweibull(20, shape=0.6) the skewness 
will be more profound.