Estimation of bivariate excess probabilities for elliptical models

Abstract

Let $(X, Y)$ be a random vector whose conditional excess probability $θ(x, y):=P(Y≤y | X>x)$ 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 $θ(x, y)$ for fixed $x, y$, with $x$ large. They respectively make use of an approximation result of Abdous et al. (cf. Canad. J. Statist. 33 (2005) 317–334, Theorem 1), a new second order refinement of Abdous et al.’s Theorem 1, and a non-approximating method. The estimation of the conditional quantile function $θ(x, ⋅)^←$ for large fixed $x$ is also addressed and these methods are compared via simulations. An illustration in the financial context is also given.