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Strong limit theorems for empirical halfspace depth trimmed regions

Andrii Ilienko, Ilya Molchanov, Riccardo Turin

Abstract

We study empirical variants of the halfspace (Tukey) depth of a probability measure $μ$, which are obtained by replacing $μ$ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of $μ$ being the uniform distribution on a convex body $K$, the depth trimmed regions are convex floating bodies of $K$, and we obtain strong limit theorems for their empirical estimators.

Strong limit theorems for empirical halfspace depth trimmed regions

Abstract

We study empirical variants of the halfspace (Tukey) depth of a probability measure , which are obtained by replacing with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of being the uniform distribution on a convex body , the depth trimmed regions are convex floating bodies of , and we obtain strong limit theorems for their empirical estimators.
Paper Structure (8 sections, 20 theorems, 130 equations, 1 figure)

This paper contains 8 sections, 20 theorems, 130 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic properties of depth trimmed regions
  3. Strong laws of large numbers
  4. Law of the iterated logarithm
  5. Distance between depth trimmed regions
  6. Strong limit theorems in the Hausdorff distance
  7. Examples
  8. Appendix: Depth functions of convex bodies

Key Result

Lemma 2.1

Under (C1), (C2), and for each $\alpha\in(0,\alpha_{\max})$,

Figures (1)

  • Figure 1: Boundaries of $\mathcal{R}(\alpha)$ and $\mathcal{R}(\alpha+\varepsilon)$ for the uniform distribution in the unit square.

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • ...and 34 more