Perturbation Properties of Depth Regions

Abstract

Let F, G be probability distributions defined on ℝ p . We examine the behaviour of halfspace depth and halfspace regions under the perturbation \( F \to \tilde F = \left( {1 - \in } \right)F + \in G,0 < \in < 1 < - \in < 1/2 \). The halfspace depth d HS (x; F) is the minimum probability of halfspaces including x and the d-level halfspace region D HS d; F), 0 < d ≤ 1, is the set of points whose depth is not less than d. The halfspace median is the centroid of the innermost depth region. The first result is an explicit expression of the perturbed halfspace depth \( {d_H}_S\left( {x;\tilde F} \right) \) which implies the influence function of the halfspace depth values. Next we prove some set-inclusion relations between the perturbed depth region \( {D_H}_S\left( {\left( {1 - \in } \right)d;\tilde F} \right) \) and \( {D_H}_S\left( {d;F} \right),{D_H}_S\left( {d - \in /\left( {1 - \in } \right);F} \right) \). The main application is an explicit expression of the perturbed halfspace median for absolutely continuous and cent rosy mmetric distributions.