Issue |
ESAIM: PS
Volume 17, 2013
|
|
---|---|---|
Page(s) | 236 - 256 | |
DOI | https://doi.org/10.1051/ps/2011161 | |
Published online | 08 February 2013 |
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
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.
Mathematics Subject Classification: 62G05 / 62G20 / 60E05 / 91B30
Key words: Level sets / distribution function / plug-in estimation / Hausdorff distance / conditional tail expectation
© EDP Sciences, SMAI, 2013
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