Smoothed Empirical Likelihood Confidence Intervals for Quantiles

Abstract

Standard empirical likelihood confidence intervals for quantiles are identical to sign-test intervals. They have relatively large coverage error, of size $n^{-1/2}$, even though they are two-sided intervals. We show that smoothed empirical likelihood confidence intervals for quantiles have coverage error of order $n^{-1}$, and may be Bartlett-corrected to produce intervals with an error of order only $n^{-2}$. Necessary and sufficient conditions on the smoothing parameter, in order for these sizes of error to be attained, are derived. The effects of smoothing on the positions of endpoints of the intervals are analysed, and shown to be only of second order.