Approximation and estimation of very small probabilities of multivariate extreme events

This article discusses the multivariate generalisation of the GW (Generalised Weibull) and log-GW tail limits, and its application to estimation of the probability of a multivariate extreme event from a sample of iid random vectors, with the probability bounded by powers of sample size with exponents below -1. A log-GW limit is reduced to a GW limit by taking the logarithm of the random variable concerned, so only the latter is considered. Its multivariate generalisation is a Tail Large Deviation Principle (LDP), the analogue for very small probabilities of classical multivariate extreme value theory based on weak convergence of measures. After standardising the marginals to a distribution function with a Weibull tail limit, dependence is represented by a homogeneous rate function. An interesting connection exists between the tail LDP and residual tail dependence (RTD), and leads to a new limit for probabilities of a wide class of tail events, generalising a recent extension of RTD. Based on the tail LDP, simple estimators for very small probabilities of extreme events are formulated. These avoid estimation of the rate function by making use of its homogeneity, and employ marginal tail estimation and "stretching" of the data cloud following a normalisation of their sample marginals. Strong consistency of the estimators is proven.