Extremal behaviour and convergence rates for sample--based geometric quantiles and half space depths

TL;DR

This paper investigates the extremal behavior and convergence rates of empirical geometric quantiles and halfspace depths, linking these rates to the tail properties of the underlying distributions and enhancing understanding of their population counterparts.

Contribution

It establishes a connection between convergence rates and tail behavior, and clarifies properties of population geometric quantiles and halfspace depths.

Findings

Convergence rates depend on distribution tails.

Sample size influences extremal behavior.

Enhanced understanding of population geometric quantiles.

Abstract

We consider the empirical versions of geometric quantile and halfspace depth, and study their extremal behaviour as a function of the sample size. The objective of this study is to establish connection between the rates of convergence and tail behaviour of the corresponding underlying distributions. The intricate interplay between the sample size and the parameter driving the extremal behaviour forms the main result of this analysis. In the process, we also fill certain gaps in the understanding of population versions of geometric quantile and halfspace depth.