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

Yao, Qiwei ORCID: 0000-0003-2065-8486 and Polonik, Wolfgang (2002) Set-indexed conditional empirical and quantile processes based on dependent data. Journal of Multivariate Analysis, 80 (2). pp. 234-255. ISSN 0047-259X

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Identification Number: 10.1006/jmva.2001.1988


We consider a conditional empirical distribution of the form Fn(C ∣ x)=∑nt=1 ωn(Xt−x) I{Yt∈C} indexed by C∈ ℓ, where {(Xt, Yt), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ωn(Xt−x)} are kernel weights, and ℓ is a class of sets. Under the assumption on the richness of the index class ℓ 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.

Item Type: Article
Official URL:
Additional Information: © 2002 Elsevier Inc
Divisions: Statistics
Subjects: H Social Sciences > HA Statistics
Date Deposited: 23 Jun 2008 08:57
Last Modified: 03 Jan 2024 21:15
Projects: L67561, L16385, ERB CHRX-CT 940693
Funders: Engineering and Physical Sciences Research Council, EU Human Capital and Mobility Program

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