Affine-Invariant Integrated Rank-Weighted Depth: Definition, Properties and Finite Sample Analysis

Because it determines a center-outward ordering of observations in $\mathbb{R}^d$ with $d\geq 2$, the concept of statistical depth permits to define quantiles and ranks for multivariate data and use them for various statistical tasks (\textit{e.g.} inference, hypothesis testing). Whereas many depth functions have been proposed \textit{ad-hoc} in the literature since the seminal contribution of \cite{Tukey75}, not all of them possess the properties desirable to emulate the notion of quantile function for univariate probability distributions. In this paper, we propose an extension of the \textit{integrated rank-weighted} statistical depth (IRW depth in abbreviated form) originally introduced in \cite{IRW}, modified in order to satisfy the property of \textit{affine-invariance}, fulfilling thus all the four key axioms listed in the nomenclature elaborated by \cite{ZuoS00a}. The variant we propose, referred to as the Affine-Invariant IRW depth (AI-IRW in short), involves the covariance/precision matrices of the (supposedly square integrable) $d$-dimensional random vector $X$ under study, in order to take into account the directions along which $X$ is most variable to assign a depth value to any point $x\in \mathbb{R}^d$. The accuracy of the sampling version of the AI-IRW depth is investigated from a nonasymptotic 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.