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For multivariate data, Tukey's half-space depth is one of the most populardepth functions available in the literature. It is conceptually simple andsatisfies several desirable properties of depth functions. The Tukey median,the multivariate median associated with the half-space depth, is also awell-known measure of center for multivariate data with several interestingproperties. In this article, we derive and investigate some interestingproperties of half-space depth and its associated multivariate median. Theseproperties, some of which are counterintuitive, have important statisticalconsequences in multivariate analysis. We also investigate a natural extensionof Tukey's half-space depth and the related median for probabilitydistributions on any Banach space (which may be finite- orinfinite-dimensional) and prove some results that demonstrate anomalousbehavior of half-space depth in infinite-dimensional spaces.

Some intriguing properties of Tukey's half-space depth