# Estimation of bivariate excess probabilities for elliptical models

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1 DMPS - Département de médecine sociale et préventive 2 MODAL-X - Modélisation aléatoire de Paris X 3 CBE STAT - College of Business and Economics Statistics Department

Abstract : Let $X,Y$ be a random vector whose conditional excess probability $\thetax,y := PY \leq y ~ | \; X >x$ 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 $\thetax,y$, for fixed $x,y$, with $x$ large. They respectively make use of an approximation result of Abdous {\it et al.} cf. \citetTheorem 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 $\thetax, \cdot^\leftarrow$ for large fixed $x$ is also addressed, and these methods are compared via simulations.

Keywords : elliptic law Conditional excess probability asymptotic independence elliptic law.

Autor: Belkacem Abdous - Anne-Laure Fougères - Kilani Ghoudi - Philippe Soulier -

Fuente: https://hal.archives-ouvertes.fr/

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