Approximation and estimation of very small probabilities of multivariate extreme events

Abstract

This article discusses modelling of the tail of a multivariate distribution function by means of a large deviation principle (LDP), and its application to the estimation of the probability pn of a multivariate extreme event from a sample of niid random vectors, with pn?[n-t2,n-t1] for some t1>1 and t2>t1. One way to view the classical tail limits is as limits of probability ratios. In contrast, the tail LDP provides asymptotic bounds or limits for log-probability ratios. After standardising the marginals to standard exponential, tail dependence is represented by a homogeneous rate function I. Furthermore, the tail LDP can be extended to represent both dependence and marginals, the latter implying marginal log-Generalised Weibull tail limits. A connection is established between the tail LDP and residual tail dependence (or hidden regular variation) and a recent extension of it. Under a smoothness assumption, they are implied by the tail LDP. Based on the tail LDP, a simple estimator for very small probabilities of extreme events is formulated. It avoids estimation of I by making use of its homogeneity. Strong consistency in the sense of convergence of log-probability ratios is proven. Simulations and an application illustrate the difference between the classical approach and the LDP-based approach.