Let's say I have a vector $x$ on $n=250,$
(in R)
x = rnorm(250)
The quantile of $\alpha=0.01$ is :
quantile(x,0.01)
1%
-2.700463
Now is that theoretically right to estimate the bootstrap quantiles of the quantile function at any confidence level ? For example I am using the BCa method described by Efron in his book:
# setting the function to estimate
theta1 = function(x){
ql = quantile(x,0.01)
return(ql)
}
# bootstrap the data with BCa method
bca = bootstrap::bcanon(x,5000,theta1,alpha=c(0.01,(1-0.01)))
# extract the two quantiles of the function
bca$confpoints
Resulting to :
alpha bca point
[1,] 0.01 -3.086066
[2,] 0.99 -2.192914
Now, what I have took, I think, is the upper(0.99) and lower(0.01) bootstrap quantiles of the quantile function at level $alpha=0.01$.
Question 1) Am I right in the last phrase? Do I interpret correctly the output? Question 2) Am I theoretically allowed to do so ?