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Journal of Probability and Statistics
Volume 2013 (2013), Article ID 629184, 7 pages
Research Article

Depth-Based Classification for Distributions with Nonconvex Support

1Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovska 83, 186 75 Praha 8, Czech Republic
2Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacky University in Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic

Received 30 April 2013; Accepted 13 August 2013

Academic Editor: Zhidong Bai

Copyright © 2013 Daniel Hlubinka and Ondrej Vencalek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Halfspace depth became a popular nonparametric tool for statistical analysis of multivariate data during the last two decades. One of applications of data depth considered recently in literature is the classification problem. The data depth approach is used instead of the linear discriminant analysis mostly to avoid the parametric assumptions and to get better classifier for data whose distribution is not elliptically symmetric, for example, skewed data. In our paper, we suggest to use weighted version of halfspace depth rather than the halfspace depth itself in order to obtain lower misclassification rate in the case of “nonconvex” distributions. Simulations show that the results of depth-based classifiers are comparable with linear discriminant analysis for two normal populations, while for nonelliptic distributions the classifier based on weighted halfspace depth outperforms both linear discriminant analysis and classifier based on the usual (nonweighted) halfspace depth.