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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 Fn of F, we estimate L(c) by Ln(c) = {Fn(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.
@article{PS_2013__17__236_0,
author = {Di Bernardino, Elena and Lalo\"e, Thomas and Maume-Deschamps, V\'eronique and Prieur, Cl\'ementine},
title = {Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory},
journal = {ESAIM: Probability and Statistics},
pages = {236--256},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2011161},
mrnumber = {3021318},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2011161/}
}
TY - JOUR AU - Di Bernardino, Elena AU - Laloë, Thomas AU - Maume-Deschamps, Véronique AU - Prieur, Clémentine TI - Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory JO - ESAIM: Probability and Statistics PY - 2013 SP - 236 EP - 256 VL - 17 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps/2011161/ DO - 10.1051/ps/2011161 LA - en ID - PS_2013__17__236_0 ER -
%0 Journal Article %A Di Bernardino, Elena %A Laloë, Thomas %A Maume-Deschamps, Véronique %A Prieur, Clémentine %T Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory %J ESAIM: Probability and Statistics %D 2013 %P 236-256 %V 17 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps/2011161/ %R 10.1051/ps/2011161 %G en %F PS_2013__17__236_0
Di Bernardino, Elena; Laloë, Thomas; Maume-Deschamps, Véronique; Prieur, Clémentine. Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 236-256. doi: 10.1051/ps/2011161
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