Bernoulli

  • Bernoulli
  • Volume 17, Number 4 (2011), 1420-1434.

Some intriguing properties of Tukey’s half-space depth

Subhajit Dutta, Anil K. Ghosh, and Probal Chaudhuri

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Abstract

For multivariate data, Tukey’s half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties. In this article, we derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey’s half-space depth and the related median for probability distributions on any Banach space (which may be finite- or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces.

Article information

Source
Bernoulli, Volume 17, Number 4 (2011), 1420-1434.

Dates
First available in Project Euclid: 4 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1320417511

Digital Object Identifier
doi:10.3150/10-BEJ322

Mathematical Reviews number (MathSciNet)
MR2854779

Zentralblatt MATH identifier
1229.62063

Keywords
Banach space depth contours half-space median $l_p$ norm symmetric distributions

Citation

Dutta, Subhajit; Ghosh, Anil K.; Chaudhuri, Probal. Some intriguing properties of Tukey’s half-space depth. Bernoulli 17 (2011), no. 4, 1420--1434. doi:10.3150/10-BEJ322. https://projecteuclid.org/euclid.bj/1320417511


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