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Asymptotics of set-indexed conditional empirical processes based on dependent data

Polonik, Wolfgang ; Yao, Qiwei

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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.

Item Type: Working paper
Place of Publication: Heidelberg
Date Deposited: 28 Jun 2016 09:35
Date: July 1998
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 510 Mathematics
Controlled Keywords: Nadaraya-Watson-Schätzer
Uncontrolled Keywords: Bahadur-Kiefer approximation; conditional distribution; covering number; empirical process theory; generalized conditional quantile; level set; minimum volume predictor; Nadaraya-Watson regression estimator; nonlinear time series; strong mixing
Schriftenreihe ID: Beiträge zur Statistik > Reports
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