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August 2022 Multivariate ρ-quantiles: A spatial approach
Dimitri Konen, Davy Paindaveine
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Bernoulli 28(3): 1912-1934 (August 2022). DOI: 10.3150/21-BEJ1404


By substituting an Lp loss function for the L1 loss function in the optimization problem defining quantiles, one obtains Lp-quantiles that, as shown recently, dominate their classical L1-counterparts in financial risk assessment. In this work, we propose a concept of multivariate Lp-quantiles generalizing the spatial (L1-)quantiles introduced by Probal Chaudhuri (J. Amer. Statist. Assoc. 91 (1996) 862–872). Rather than restricting to power loss functions, we actually allow for a large class of convex loss functions ρ. We carefully study existence and uniqueness of the resulting ρ-quantiles, both for a general probability measure over Rd and for a spherically symmetric one. Interestingly, the results crucially depend on ρ and on the nature of the underlying probability measure. Building on an investigation of the differentiability properties of the objective function defining ρ-quantiles, we introduce a companion concept of spatial ρ-depth, that generalizes the classical spatial depth. We study extreme ρ-quantiles and show in particular that extreme Lp-quantiles behave in fundamentally different ways for p2 and p>2. Finally, we establish Bahadur representation results for sample ρ-quantiles and derive their asymptotic distributions. Throughout, we impose only very mild assumptions on the underlying probability measure, and in particular we never assume absolute continuity with respect to the Lebesgue measure.

Funding Statement

Research is supported by the Program of Concerted Research Actions (ARC) of the Université libre de Bruxelles and by an Aspirant fellowship from the FNRS (Fonds National pour la Recherche Scientifique), Communauté Française de Belgique.


The authors would like to thank the Editor-In-Chief, Mark Podolskij, the Associate Editor, and two anonymous referees for their insightful comments and suggestions, as well as Gilles Stupfler for inspiring discussions on this work and its possible extensions.


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Dimitri Konen. Davy Paindaveine. "Multivariate ρ-quantiles: A spatial approach." Bernoulli 28 (3) 1912 - 1934, August 2022.


Received: 1 March 2021; Published: August 2022
First available in Project Euclid: 25 April 2022

Digital Object Identifier: 10.3150/21-BEJ1404

Keywords: Bahadur representation results , convex objective functions , M-estimation , Multivariate quantiles , spatial depth , spatial quantiles


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Vol.28 • No. 3 • August 2022
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