Some intriguing properties of Tukey’s half-space depth

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. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2011, Vol. 17, No. 4, 1420-1434. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2011 ISI/BSTukey's half-space depth 1421 infinity. Moreover, if the underlying population distribution F has a spherically symmetric density f , that is, f (x) = ψ( x 2 ) for some ψ : R + → R + , the half-space depth turns out to be a decreasing function of x 2 = (|x 1 | 2 + · · · + |x d | 2 ) 1/2 . Consequently, when ψ is monotonically decreasing (i.e., f is unimodal), the half-space depth becomes an increasing function of f and vice versa. Therefore, in such cases, the half-space depth contours coincide with the contours of the density function. Because of this property of the halfspace depth, classification rules based on the ordering of the half-space depth functions coincide with the optimal Bayes classifier for discriminating among spherically symmetric unimodal populations differing in their centers of symmetry (see, e.g., [8]). Similarly, the use of the half-space depth functions to order and trim multivariate data sets (see, e.g., [6,17]) leading to the determination of central and outlying observations has a natural justification when the density contours coincide with the half-space depth contours. Also, due to this relation between half-space depth and spherical symmetry, half-space depth has been used to construct diagnostic tools for checking spherical symmetry of a data cloud (see, e.g., [13], pages 809-811). Another well-known feature of half-space depth is its characterization property. Koshevoy [10] proved that if the half-space depth functions of two atomic measures with finite support are identical, then the measures are also identical. Cuesta-Albertosa and Nieto-Reyes [4] proved this characterization property of Tukey depth for discrete distributions. Under some regularity conditions, Koshevoy [11] proved this characterization property for absolutely continuous probability distributions with compact support in finite-dimensional spaces. Hassairi and Regaieg [9] ge...

From Depth to Local Depth: A Focus on Centrality

“…x 0 is distribution-free under the null, which allows to approximate arbitrary well the exact fixedn critical values through simulations; • the test from Dutta, Ghosh, and Chaudhuri (2011) that may be seen as the companion test for the null of angular symmetry about an unspecified center, as it rejects this null for large values of T (n) = T (n) θ θ θ , whereθ θ θ denotes the halfspace deepest point of P (n) (or, if unicity fails, the barycenter of the collection of deepest points). Critical values are obtained from bootstrap-type samples (as in Dutta, Ghosh, and Chaudhuri (2011), we will use the term "bootstrap," although the corresponding tests are rather of a permutation nature).…”

On smoothness of Tukey depth contours

“…Provided, in addition, the smoothness of C (such as p ∈ (1, ∞) for L p symmetrical distributions), f itself is a smooth function on the whole space R 2 . Dutta et al (2011) stated that for p = 2 the Tukey depth contours coincide with the contours of an L 2 norm, no matter of what form ρ is. Additionally, in Section 2 the authors prove that for any p = 2 the contours of the density cannot coincide with the Tukey depth contours for ρ decreasing.…”

“…To this end, we investigate mixtures of Gaussian distributions, L p symmetrical distributions (in line with the research of Dutta et al (2011)), and finally the class of (strictly) quasi-concave distributions with smooth densities.…”

Half-space mass: a maximally robust and efficient data depth method

“…It is a maximally robust estimator with properties given in the following theorem: Dutta et al (2011) reveal that, for a size n dataset in a d > n dimensional space, since the d-dimensional volume of the convex hull of such dataset is going to be zero, half-space depth will behave anomalously having 0 measures almost everywhere in d . In such cases, half-space depth does not carry any useful statistical information.…”

“…Since then it has been extensively studied. Donoho and Gasko (1992) have revealed the breakdown point of Tukey median; Zuo and Serfling (2000) have compared it to various competitors and Dutta et al (2011) have investigated the properties of half-space depth. Meanwhile, the concept of data depth has been adopted for multivariate statistical analysis since it provides a nonparametric approach that does not rely on the assumption of normality (Liu et al 1999).…”