Asymptotics of conditional empirical processes
Journal of Multivariate Analysis
A smooth conditional quantile estimator and related applications of conditional empirical processes
Journal of Multivariate Analysis
Density estimation under qualitative assumptions in higher dimensions
Journal of Multivariate Analysis
A kernel estimator of a conditional quantile
Journal of Multivariate Analysis
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We consider a conditional empirical distribution of the form Fn(C | x) = Σt = 1n ωn(Xt - x) I{Yt ∈ C} indexed by C ∈ b, where {(Xt, Yt), t = 1, ..., n} are observations from a strictly stationary and strong mixing stochastic process, {ωn(Xt- x)} are kernel weights, and b is a class of sets. Under the assumption on the richness of the index class b in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for Fn(. |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.