Set-indexed conditional empirical and quantile processes based on dependent data

Abstract

We consider a conditional empirical distribution of the form F̂n(C x) = ∑tn=1 ωn(Xt-x) I{YtεC} indexed by CεC, where {(Xt, Yt), t = 1, ..., n} are observations from a strictly stationary and strong mixing stochastic process, {ωn(Xt-x)} are kernel weights, and C is a class of sets. Under the assumption on the richness of the index class C in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for F̂n(· x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur-Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed. © 2001 Elsevier Science.