Data depth for the uniform distribution

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Abstract

Given a set X of k points and a point z in the n-dimensional euclidean space, the Tukey depth of z with respect to X, is defined as m/k, where m is the minimum integer such that z is not in the convex hull of some set of k-m points of X. If z belongs to the closed region B delimited by an ellipsoid, define the continuous depth of z with respect to B as the quotient V(z)/Vol(B), where V(z) is the minimum volume of the intersection of B with the halfspaces defined by any hyperplane passing through z, and Vol(B) is the volume of B. We consider z a random variable and prove that, if z is uniformly distributed in B, the continuous depth of z with respect to B has expected value 1/2^n+1. This result implies that if z and X are uniformly distributed in B, the expected value of Tukey depth of z with respect to X converges to 1/2^n+1 as the number of points k goes to infinity. These findings have applications in ecology, namely within the niche theory, where it is useful to explore and characterize the distribution of points inside species niche.
Original languageUnknown
Pages (from-to)27-39
JournalEnvironmental And Ecological Statistics
Volume21
Issue number1
DOIs
Publication statusPublished - 1 Jan 2014

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