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Minimum Volume Sets and Generalized Quantile Processes

Polonik, Wolfgang

In: Stochastic processes and their applications, 69 (1997), Nr. 1. pp. 1-24. ISSN 0304-4149

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Abstract

Minimum volume sets in classes C of subsets of the d-dimensionalEuclidean space can be used as estimators of level sets of a density. By usingempirical process theory consistency results and rates of convergence forminimum volume sets are given which depend on entropy conditions on C .The volume of the minimum volume sets itself, which can be used for robustestimation of scale, can be considered as a generalized quantile process inthe sense of Einmahl and Mason (1992). Bahadur-Kiefer approximations forgeneralized quantile processes are given which generalize classical resultson the one-dimensional quantile process. Rates of convergence of minimumvolume sets can be used to obtain Bahadur-Kiefer approximations and viceversa. A generalization of the minimum volume approach to regressionproblems and spectral analysis is presented.

Item Type: Article
Journal or Publication Title: Stochastic processes and their applications
Volume: 69
Number: 1
Publisher: Elsevier
Place of Publication: Amsterdam
Date Deposited: 16 Jun 2016 07:39
Date: 1997
ISSN: 0304-4149
Page Range: pp. 1-24
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 510 Mathematics
Uncontrolled Keywords: Bahadur-Kiefer approximation; Empirical process theory; Generalized uniform empirical process; Level set estimation; Rates of convergence; Test for multimodality
Schriftenreihe ID: Beiträge zur Statistik > Beiträge
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