Open Access
2022 Depth level set estimation and associated risk measures
Sara Armaut, Roland Diel, Thomas Laloë
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Electron. J. Statist. 16(2): 6584-6630 (2022). DOI: 10.1214/22-EJS2095


Depth functions have become increasingly powerful tools in non-parametric inference for multivariate data as they measure a degree of centrality of a point with respect to a distribution. A multivariate risk scenario is then represented by a depth-based lower level set of the risk factors, meaning that we consider a non-compact setting. The aim of this paper is to study the asymptotic behavior of level sets of a general multivariate depth function and a particular multivariate risk measure, the Covariate-Conditional-Tail-Expectation (CCTE) based on a depth function. More precisely, given a probability measure P on Rd and a depth function D(,P), we are interested in the α-lower level set LD(α):=zRd:D(z,P)α. First, we present a plug-in approach in order to estimate LD(α), then we derive consistency of its estimator under some regularity conditions. In a second part, we provide a consistent estimator of the CCTE for a general depth function with a rate of convergence and we consider the particular case of Mahalanobis depth. Finally, a simulation study complements the performances of our estimator and an application on real data is presented.

Funding Statement

This work has been supported by the project ANR McLaren (ANR-20-CE23-0011).


The authors express their gratitude to two anonymous Referees and Associate Editor for their valuable comments on this article.


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Sara Armaut. Roland Diel. Thomas Laloë. "Depth level set estimation and associated risk measures." Electron. J. Statist. 16 (2) 6584 - 6630, 2022.


Received: 1 March 2022; Published: 2022
First available in Project Euclid: 19 December 2022

MathSciNet: MR4522941
zbMATH: 07650530
Digital Object Identifier: 10.1214/22-EJS2095

Primary: 60G32 , 62G05 , 62H12

Keywords: level set estimation , Mahalanobis depth , Multivariate depth function , plug-in , risk measure

Vol.16 • No. 2 • 2022
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