Smooth quantile estimators under strong mixing: necessary and sufficient conditions on bandwidth for weak convergence

Abstract

We study the smooth quantile estimator Q ̂ (p), 0<p<1 based on a kernel k and a sequence of bandwidth a n >0 for a sequence of stationary strong mixing random variables. Under minimal assumptions on the underlying distribution function F and kernel k, we establish necessary and sufficient conditions on a n for the Central Limit Theorem to hold for Q ̂ (p), 0<p<1 . Our results extend the Central Limit Theorems of Ralescu and Sun (J. Statist. Plann. Inference 35 (1993) 55) and Sen (J. Multivariate Anal. 2 (1972) 77).