On the Distribution of a Studentized Quantile

SUMMARY The asymptotic variance of a sample quantile depends on the value of the population density at the population quantile. Therefore Studentizing a sample quantile involves density estimation, either explicitly or implicitly. One popular way of Studentizing is to use the Siddiqui–Bloch–Gastwirth estimator, whose construction depends crucially on the choice of a smoothing parameter m. We examine the effect which the selection of m has on the level error of tests or confidence intervals based on Studentized quantiles and show that, if we wish to minimize this error, m should be of a smaller order of magnitude than is recommended by squared error theory. The cases of one- and two-sided procedures are distinctly different, the former being less sensitive to the choice of m.