Q statistics in data depth: fundamental theory revisited and variants

TL;DR

This paper revisits the theoretical foundations of the Q statistic in data depth, improving its asymptotic expansion rate and enabling the development of more powerful tests and variants.

Contribution

It introduces new assumptions and proof methods to achieve the optimal asymptotic rate for the Q statistic, strengthening its theoretical basis and potential applications.

Findings

Achieved the target asymptotic rate of m^{-1} for the Q statistic.

Revised assumptions enable higher-order expansions for more powerful tests.

Opened avenues for developing new variants and applications of data depth methods.

Abstract

Recently, data depth has been widely used to rank multivariate data. The study of the depth-based $Q$ statistic, originally proposed by Liu and Singh (1993), has become increasingly popular when it can be used as a quality index to differentiate between two samples. Based on the existing theoretical foundations, more and more variants have been developed for increasing power in the two sample test. However, the asymptotic expansion of the $Q$ statistic in the important foundation work of Zuo and He (2006) currently has an optimal rate $m^{-3/4}$ slower than the target $m^{-1}$, leading to limitations in higher-order expansions for developing more powerful tests. We revisit the existing assumptions and add two new plausible assumptions to obtain the target rate by applying a new proof method based on the Hoeffding decomposition and the Cox-Reid expansion. The aim of this paper is to rekindle interest in asymptotic data depth theory, to place Q-statistical inference on a firmer theoretical basis, to show its variants in current research, to open the door to the development of new theories for further variants requiring higher-order expansions, and to explore more of its potential applications.