Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extreme Geometric Quantiles Under Minimal Assumptions, with a Connection to Tukey Depth

Sibsankar Singha, Marie Kratz, Sreekar Vadlamani

Abstract

Geometric (also known as spatial) quantiles, introduced by Chaudhury and representing one of the three principal approaches to defining multivariate quantiles, have been well studied in the literature. In this work, we focus on the extremal behaviour of these quantiles. We establish new extremal properties, namely general lower and upper bounds for the norm of extreme geometric quantiles, free of any moment conditions. We discuss the impact of such results on the characterization of distribution behaviour. Importantly, the lower bound can be directly linked to univariate quantiles and to halfspace (Tukey) depth central regions, highlighting a novel connection between these two fundamental notions of multivariate quantiles.

Extreme Geometric Quantiles Under Minimal Assumptions, with a Connection to Tukey Depth

Abstract

Geometric (also known as spatial) quantiles, introduced by Chaudhury and representing one of the three principal approaches to defining multivariate quantiles, have been well studied in the literature. In this work, we focus on the extremal behaviour of these quantiles. We establish new extremal properties, namely general lower and upper bounds for the norm of extreme geometric quantiles, free of any moment conditions. We discuss the impact of such results on the characterization of distribution behaviour. Importantly, the lower bound can be directly linked to univariate quantiles and to halfspace (Tukey) depth central regions, highlighting a novel connection between these two fundamental notions of multivariate quantiles.
Paper Structure (7 sections, 7 theorems, 83 equations, 1 figure)

This paper contains 7 sections, 7 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General lower and upper bounds for the extremal geometric quantile norm
  4. Under integrability conditions
  5. Conclusion
  6. Geometric Quantile versus Halfspace Depth
  7. Rate of Convergence for the Direction of Geometric Quantiles

Key Result

Theorem 3.1

Let $u \in S^{d-1}$ and $0<\alpha < 1$.

Figures (1)

  • Figure 1: Depth contours for different bivariate distributions

Theorems & Definitions (15)

  • Definition 2.1: Geometric quantile Chaudhury1996
  • Definition 2.2: Halfspace depth; Tukey1975
  • Theorem 3.1: Upper bound
  • Corollary 3.2
  • Theorem 3.3
  • Remark 3.4: On the geometric constant $M_\gamma$
  • Remark 3.5: On the admissible range of $\alpha$
  • Proposition 3.6
  • Theorem 3.7
  • Remark 3.8
  • ...and 5 more