# Estimation of bivariate excess probabilities for elliptical models

Abstract : Let $(X,Y)$ be a random vector whose conditional excess probability $\theta(x,y) := P(Y \leq y ~ | \; X >x)$ is of interest. Estimating this kind of probability is a delicate problem as soon as $x$ tends to be large, since the conditioning event becomes an extreme set. Assume that $(X,Y)$ is elliptically distributed, with a rapidly varying radial component. In this paper, three statistical procedures are proposed to estimate $\theta(x,y)$, for fixed $x,y$, with $x$ large. They respectively make use of an approximation result of Abdous {\it et al.} (cf. \citet[Theorem 1]{AFG05}), a new second-order refinement of Abdous {\it et al.}'s Theorem 1, and a non-approximating method. The estimation of the conditional quantile function $\theta(x, \cdot)^\leftarrow$ for large fixed $x$ is also addressed, and these methods are compared via simulations.
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Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2008, 14 (4), pp.1065-1088. 〈10.3150/08-BEJ140〉
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https://hal.archives-ouvertes.fr/hal-00117001
Contributeur : Philippe Soulier <>
Soumis le : jeudi 24 avril 2008 - 17:02:54
Dernière modification le : mardi 23 janvier 2018 - 16:00:02
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 17:31:32

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Belkacem Abdous, Anne-Laure Fougères, Kilani Ghoudi, Philippe Soulier. Estimation of bivariate excess probabilities for elliptical models. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2008, 14 (4), pp.1065-1088. 〈10.3150/08-BEJ140〉. 〈hal-00117001v3〉

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