Multivariate ρ-quantiles: A spatial approach

Abstract

By substituting an $L_p$ loss function for the $L_1$ loss function in the optimization problem defining quantiles, one obtains $L_p$-quantiles that, as shown recently, dominate their classical $L_1$-counterparts in financial risk assessment. In this work, we propose a concept of multivariate $L_p$-quantiles generalizing the spatial ($L_1$-)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 ${\mathbb{R}}^d$ 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 $L_p$-quantiles behave in fundamentally different ways for $p\le 2$ 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.