HAL: hal-00117001, version 2

arXiv: math/0611914

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v1 (2006-11-29)

v2 (2007-07-28)

v3 (2008-04-24)

Estimation of bivariate excess probabilities for elliptical models

Belkacem Abdous 1, Anne-Laure Fougères 2, Kilani Ghoudi 3, Philippe Soulier 2

(2006-11-29)

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.

1:	Département de médecine sociale et préventive (DMPS)
	Université Laval
2:	Modélisation aléatoire de Paris X (MODAL'X)
	Université Paris X - Paris Ouest Nanterre La Défense
3:	College of Business and Economics Statistics Department (CBE STAT)
	United Arab Emirates University

Subject

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Mathematics/Statistics

Keyword(s): Conditional excess probability – asymptotic independence – elliptic law.

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From: Philippe Soulier <>

Submitted on: Saturday, 28 July 2007 00:15:07

Updated on: Saturday, 28 July 2007 07:53:51