%0 Journal Article
%@ 0304-4149
%A Polonik, Wolfgang
%C Amsterdam
%D 1997
%F heidok:21331
%I Elsevier
%J Stochastic processes and their applications
%K Bahadur-Kiefer approximation; Empirical process theory;   Generalized uniform empirical process; Level set estimation; Rates of convergence; Test for multimodality
%N 1
%P 1-24
%R 10.11588/heidok.00021331
%T Minimum Volume Sets and Generalized Quantile Processes
%U https://archiv.ub.uni-heidelberg.de/volltextserver/21331/
%V 69
%X 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.