Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Abstract

This paper deals with the problem of estimating the level sets L(c) = |F(x) ≥ c|, with c ε (0, 1), of an unknown distribution function F on ℝ2+. A plug-in approach is followed. That is, given a consistent estimator Fn of F, we estimate L(c) by Ln(c) = |Fn(x) ≥ c|. In our setting, noncompactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In particular we propose a new bivariate version of the conditional tail expectation by conditioning the two-dimensional random vector to be in the level set L(c). We also present simulated and real examples which illustrate our theoretical results. © EDP Sciences, SMAI 2013.