A0251
Title: Affine-invariant integrated rank-weighted depth: Definition, properties and finite sample analysis
Authors: Guillaume Staerman - Inria, Universite Paris-Saclay (France) [presenting]
Abstract: Whereas many depth functions have been proposed ad-hoc in the literature since the seminal contribution, not all of them possess the properties desirable to emulate the notion of quantile function for univariate probability distributions. We propose an extension of the integrated rank-weighted (IRW) statistical depth, modified in order to satisfy the property of affine-invariance, fulfilling thus all the four key axioms. The variant we propose, referred to as the Affine-Invariant IRW depth (AI-IRW in short), involves the precision matrix of the d-dimensional random vector $X$ under study, in order to take into account the directions along which $X$ is the most variable to assign a depth value to any point $x$ in $R^d$. The accuracy of the sampling version of the AI-IRW depth is investigated from a non-asymptotic perspective. Namely, a concentration result for the statistical counterpart of the AI-IRW depth is proved. Beyond the theoretical analysis carried out, applications to anomaly detection are considered, and numerical results are displayed, providing strong empirical evidence of the relevance of the depth function we propose here.
