  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 (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.
