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2023 Affine invariant integrated rank-weighted statistical depth: properties and finite sample analysis
Stephan Clémençon, Pavlo Mozharovskyi, Guillaume Staerman
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Electron. J. Statist. 17(2): 3854-3892 (2023). DOI: 10.1214/23-EJS2189


Because it determines a center-outward ordering of observations in Rd with d2, 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 ad-hoc in the literature since the seminal contribution of [50], 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 integrated rank-weighted statistical depth (IRW depth in abbreviated form) originally introduced in [40], modified in order to satisfy the property of affine invariance, fulfilling thus all the four key axioms listed in the nomenclature elaborated by [59]. The variant we propose, referred to as the affine invariant IRW depth (AI-IRW in short), involves the precision matrix 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 xRd. 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.

Funding Statement

G.S. was supported by BPI France in the context of the PSPC Project Expresso (2017-2021) and the industrial chair DSAIDIS of Télécom Paris.


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Stephan Clémençon. Pavlo Mozharovskyi. Guillaume Staerman. "Affine invariant integrated rank-weighted statistical depth: properties and finite sample analysis." Electron. J. Statist. 17 (2) 3854 - 3892, 2023.


Received: 1 November 2022; Published: 2023
First available in Project Euclid: 11 December 2023

Digital Object Identifier: 10.1214/23-EJS2189

Primary: 62G05
Secondary: 62G30 , 62H99 , 68Q32

Keywords: affine invariance , anomaly detection , Concentration inequalities , integrated rank-weighted depth , Statistical depth

Vol.17 • No. 2 • 2023
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