Concentration of the empirical level sets of Tukey’s halfspace depth

Abstract

Tukey's halfaspace depth has attracted much interest in data analysis, because it is a natural way of measuring the notion of depth relative to a cloud of points or, more generally, to a probability measure. Given an i.i.d. sample, we investigate the concentration of upper level sets of the Tukey depth relative to that sample around their population version. We show that under some mild assumptions on the underlying probability measure, concentration occurs at a parametric rate and we deduce moment inequalities at that same rate. In a computational prospective, we study the concentration of a discretized version of the empirical upper level sets.

Halfspace depth and floating body

“…In Theorem 2, (P n ) δ stands for the δ-central region (7) of the empirical measure P n . Further valuable improvements of the statistical theory of the halfspace depth include the derivation of the rates of convergence of the depth and its central regions [84,29,28], and distributional asymptotics of these and related quantities [8,185,186,115,116,117] (9) can be found in literature much earlier than the definition of the halfspace depth (see [151,Sections 3 and 4]). From these references, it appears that the behavior of the maximal depth relates to the degree of concavity of the measure P .…”

“…(Convex) floating bodies for general measures have already been considered in the literature, mainly due to the association of convex bodies and log-concave measures established by Ball [10]. The previous definitions were considered by Werner [179], Bobkov [18], Fresen [57,58], and Brunel [28], among others. In connection with the halfspace depth, the floating bodies (27) were considered in Nolan [134], and Massé and Theodorescu [118].…”

“…The (Dupin's) floating body P [δ] of a general measure P may not exist. Sufficient conditions for the existence of floating bodies of probability measures appear to be a challenging problem of great importance in mathematical statistics, and the theory of data depth (see, e.g., [28,Open question 1], or [116,117]). Many theoretical results on the behavior of the depth and its central regions hold true only under the assumption of existence of floating bodies of P , see also the discussion in Section 8 below.…”

Methods for Estimation of Convex Sets

“…Multivariate quantile estimation: level sets of the Tukey depth [Bru18a] As already pointed out in [FJ66,Fis69,FV14], the study of the convex hull of a cloud of points is a multivariate extension of extreme value theory. Indeed, by Lemma 7, the support function ofK n in any direction u ∈ S d−1 is given by the maximum of i.i.d.…”

“…In general, neither u ↦ q u nor u ↦q u are subadditive, and hence, they are not the support functions of the sets G andĜ n respectively. However, u ↦q u is always continuous, as a consequence of [Bru18a, Lemma 15] and u ↦ q u is continuous under some weak assumptions on µ (see [Bru18a,Lemma 11]). Building on standard results from empirical process theory, [Bru18a] shows that under some assumption on µ, sup u∈S d−1 q u − q u is small, with high probability.…”