Some intriguing properties of Tukey's half-space depth

Details Some intriguing properties of Tukey's half-space depth Some intriguing properties of Tukey's half-space depth

2012-01-05

arxiv.org/abs/1201.1171v1

Bernoulli 2011, Vol. 17, No. 4, 1420-1434

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ322 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ322

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.

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