  The aim of this paper is to study the asymptotic behavior of a particular
multivariate risk measure, the Covariate-Conditional-Tail-Expectation (CCTE),
based on a multivariate statistical depth function. Depth functions have become
increasingly powerful tools in nonparametric inference for multivariate data,
as they measure a degree of centrality of a point with respect to a
distribution. A multivariate risks scenario is then represented by a
depth-based lower level set of the risk factors, meaning that we consider a
non-compact setting. More precisely, given a multivariate depth function D
associated to a fixed probability measure, we are interested in the lower level
set based on D. First, we present a plug-in approach in order to estimate the
depth-based level set. In a second part, we provide a consistent estimator of
our CCTE for a general depth function with a rate of convergence, and we
consider the particular case of the Mahalanobis depth. A simulation study
complements the performances of our estimator.
