Wolfgang Polonik, Universitaet Heidelberg
Qiwei Yao, University of Kent at Canterbury
Abstract
We study the asymptotic properties of the a conditional empirical process defined from weakly dependent observations indexed by a class of sets. Under the assumption on the richness of the index class in terms of metric entropy with bracketing, we have established uniform convergence, and asymptotic normality for the conditional empirical process. The key technical result gives rates of convergences for the sup-norm of the conditional empirical process over a sequence of set claseese with decreasing maximum L1-norm. The results are then applied to derive Bahadur-Kiefer type approximations for a generalized conditional quantile process which is closely related to the minimum volume sets. The potential applications in the areas of estimation of level sets and testing for unimodality of conditional distributions are discussed.