Estimation of bivariate excess probabilities for elliptical models

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.