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Abstract

Based on observation vectors (Xt,Yt) from a strong mixing stochastic process,we estimate the conditional distribution of Y given X = x by means of aNadaraya-Watson-type estimator. Using this, we study the asymptotics ofa conditional empirical process indexed by classes of sets.Under assumptions on the richness of the indexing class in terms ofmetric entropy with bracketing, we have established uniform convergence, and asymptotic normality. The key technical result gives rates of convergences for the sup-norm of the conditional empirical process over a sequenceof indexing classes with decreasing maximum Lt-norm.The results are then applied to derive Bahadur-Kiefer type approximationsfor a generalized conditional quantile process which is closelyrelated to the minimum volume sets. The potential applications in the areas ofestimation of level sets and testing for unimodality of conditionaldistributions are discussed.