# Bootstrap – Analysis of BCa Quantiles of the Quantile Function

bootstrapquantilessimulation

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 ?