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

• Volume: 17, page 236-256
• ISSN: 1292-8100

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## Abstract

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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.

## How to cite

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Di Bernardino, Elena, et al. "Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory." ESAIM: Probability and Statistics 17 (2013): 236-256. <http://eudml.org/doc/274392>.

@article{DiBernardino2013,
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 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.},
author = {Di Bernardino, Elena, Laloë, Thomas, Maume-Deschamps, Véronique, Prieur, Clémentine},
journal = {ESAIM: Probability and Statistics},
keywords = {level sets; distribution function; plug-in estimation; Hausdorff distance; conditional tail expectation},
language = {eng},
pages = {236-256},
publisher = {EDP-Sciences},
title = {Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory},
url = {http://eudml.org/doc/274392},
volume = {17},
year = {2013},
}

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
PB - EDP-Sciences
VL - 17
SP - 236
EP - 256
AB - 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.
LA - eng
KW - level sets; distribution function; plug-in estimation; Hausdorff distance; conditional tail expectation
UR - http://eudml.org/doc/274392
ER -

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