Universität zu Köln

Nagy, Stanislav, Dyckerhoff, Rainer and Mozharovskyi, Pavlo (2020). Uniform convergence rates for the approximated halfspace and projection depth. Electron. J. Stat., 14 (2). S. 3939 - 3976. CLEVELAND: INST MATHEMATICAL STATISTICS. ISSN 1935-7524

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Abstract

The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Nagy, Stanislav UNSPECIFIED UNSPECIFIED UNSPECIFIED Dyckerhoff, Rainer UNSPECIFIED UNSPECIFIED UNSPECIFIED Mozharovskyi, Pavlo UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-349678
DOI:	10.1214/20-EJS1759
Journal or Publication Title:	Electron. J. Stat.
Volume:	14
Number:	2
Page Range:	S. 3939 - 3976
Date:	2020
Publisher:	INST MATHEMATICAL STATISTICS
Place of Publication:	CLEVELAND
ISSN:	1935-7524
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language MAXIMAL-SPACINGS; ASYMPTOTICS; STATISTICS; LAWS Multiple languages Statistics & Probability Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/34967