A large deviations approach to the statistics of extreme events

A large deviations approach to the statistics of extreme events addresses the statistical analysis of extreme events with very low probabilities: given a random sample of data of size n, the probability is much smaller than 1/n. In particular, it takes a close look at the regularity assumptions on the tail of the (univariate or multivariate) distribution function. The classical assumptions, cast in the form of limits of ratios of probabilities of extreme events, are not directly applicable in this setting. Therefore, additional assumptions are commonly imposed. Because these may be very restrictive, this thesis proposes an alternative regularity assumption, taking the form of asymptotic bounds on ratios of logarithms of probabilities of extreme events, i.e., a large deviation principle (LDP). In the univariate case, this tail LDP is equivalent to the log-Generalised Weibull (log-GW) tail limit, which generalises the Weibull tail limit and the classical Pareto tail limit, amongst others. Its application to the estimation of high quantiles is discussed. In the multivariate case, the tail LDP implies marginal log-GW tail limits together with a standardised tail LDP describing tail dependence. Its application to the estimation of very low probabilities of multivariate extreme events is discussed, and a connection is established to hidden regular variation (residual tail dependence) and similar models.