TY - JOUR

T1 - Data depth for the uniform distribution

AU - Cerdeira, Jorge Orestes Lasbarrères

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Hyperspherical cap

KW - mean value

KW - uniform distribution

KW - Tukey depth

KW - species niche

U2 - 10.1007/s10651-013-0242-7

DO - 10.1007/s10651-013-0242-7

M3 - Article

SN - 1352-8505

VL - 21

SP - 27

EP - 39

JO - Environmental And Ecological Statistics

JF - Environmental And Ecological Statistics

IS - 1

ER -