Depth level set estimation and associated risk measures

Abstract

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 ${\mathbb{R}}^d$ and a depth function $D(\cdot ,P)$, we are interested in the α-lower level set $L_D(\mathit{\alpha }):=\left\{z\in {\mathbb{R}}^d:D(z,P)\le \mathit{\alpha }\right\}$. First, we present a plug-in approach in order to estimate $L_D(\mathit{\alpha })$, 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.