Some intriguing properties of Tukey’s half-space depth. (English) Zbl 1229.62063

Summary: For multivariate data, J. W. Tukey’s [Proc. Int. Congr. Math., Vancouver 1974, Vol. 2, 523–531 (1975; Zbl 0347.62002)] 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.
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.


62H05 Characterization and structure theory for multivariate probability distributions; copulas
62A09 Graphical methods in statistics
46N30 Applications of functional analysis in probability theory and statistics


Zbl 0347.62002
Full Text: DOI arXiv


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