Estimation of bivariate excess probabilities for elliptical models
Résumé
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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