Bootstrap Confidence Interval for a Quantile

quantile.CI.via.bootstrap <- function(x, p, alpha = 0.1) {
    ## Purpose:
    ##   Calculate a two-sided confidence interval with confidence level of (1 - alpha) for
    ##   a quantile, based on the (computing intensive) bootstrap resampling method. 
    ##
    ## Arguments:
    ##   - x: a vector of values, representing a data sample. 
    ##   - p: probability cutpoint for the quantile (between 0 and 1).
    ##   - alpha: type I error level (default to 0.1 so a 90% CI is calculated)
    ##
    ## Return:
    ##   - CI: the lower and upper limits of the two-sided CI. 

    q <- quantile(x, probs = p)         
    message("Quantile Point Estimate = ", q, " (Probability Cutpoint = ", p, ")\n")

    ## Bootstrap resampling with 2000 replications
    library(boot)
    set.seed(1)
    b <- boot(x, function(x, i) quantile(x[i], probs = p), R = 2000)

    boot.ci(b, conf = 1 - alpha, type = c("norm", "basic", "perc", "bca"))

}
if (F) {                                # Unit Test
    x <- 1:100
    p <- 0.9
    alpha <- 0.05
    quantile.CI.via.bootstrap(x, p, alpha)
    ## Intervals : 
    ## Level      Normal              Basic         
    ## 95%   (84.50, 96.34 )   (85.10, 97.10 )  

    ## Level     Percentile            BCa          
    ## 95%   (83.1, 95.1 )   (83.3, 95.1 )  
}