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

Published online by Cambridge University Press:  08 February 2013

Elena Di Bernardino ,
Thomas Laloë ,
Véronique Maume-Deschamps  and
Clémentine Prieur
Elena Di Bernardino
Affiliation:
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
Thomas Laloë
Affiliation:
Université de Nice Sophia-Antipolis, Laboratoire J-A Dieudonné, Parc Valrose, 06108 Nice Cedex 02, France; thomas.laloe@unice.fr
Véronique Maume-Deschamps
Affiliation:
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
Clémentine Prieur
Affiliation:
Université Joseph Fourier, Tour IRMA, MOISE-LJK B.P. 53 38041 Grenoble, France; clementine.prieur@imag.fr
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Abstract

This paper deals with the problem of estimating the level setsL(c) =  {F(x) ≥ c},with c ∈ (0,1), of an unknown distribution function F on ℝ+2. A plug-inapproach is followed. That is, given a consistent estimatorFn of F, we estimateL(c) byLn(c) =  {Fn(x) ≥ c}.In our setting, non-compactness property is a priori required for thelevel sets to estimate. We state consistency results with respect to the Hausdorffdistance and the volume of the symmetric difference. Our results are motivated byapplications in multivariate risk theory. In particular we propose a new bivariate versionof the conditional tail expectation by conditioning the two-dimensional random vector tobe in the level set L(c). We also present simulated andreal examples which illustrate our theoretical results.

Keywords

Level setsdistribution functionplug-in estimationHausdorff distanceconditional tail expectation

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 236 - 256
Copyright
© EDP Sciences, SMAI, 2013

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