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

¹ Universitéde Lyon, Université Lyon 1, ISFA, Laboratoire SAF, 50 Avenue Tony Garnier, 69366 Lyon, France
elena.di-bernardino@univ-lyon1.fr; veronique.maume@univ-lyon1.fr
² Université de Nice Sophia-Antipolis, Laboratoire J-A Dieudonné, Parc Valrose, 06108 Nice Cedex 02, France
thomas.laloe@unice.fr
³ Université Joseph Fourier, Tour IRMA, MOISE-LJK B.P. 53 38041 Grenoble, France
clementine.prieur@imag.fr

Received: 22 March 2011
Revised: 13 July 2011

Abstract

This paper deals with the problem of estimating the level sets L(c) =  {F(x) ≥ c}, with c ∈ (0,1), of an unknown distribution function F on ℝ₊². A plug-in approach is followed. That is, given a consistent estimator F_n of F, we estimate L(c) by L_n(c) =  {F_n(x) ≥ c}. In our setting, non-compactness 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.