Exceedance Over Threshold

Abstract

We discuss how to fit threshold models to the exceedances over high thresholds. In a univariate setting, often the crucial step is to estimate the extreme value index, since the accuracy achieved there essentially determines the estimation error for extreme quantiles or exceedance probabilities. In this context, the choice of the sample fraction used for estimation is particularly important. Besides classical models with independent and identically distributed observations, it is also discussed how to cope with serial dependence, trends and seasonality. When fitting a multivariate threshold model, one has to estimate whole functions that describe the tail dependence structure. Here mainly semiparametric and nonparametric estimators for the stable tail dependence function, the exponent measure and the angular measure are considered. In addition, we briefly study the problem of testing for asymptotic independence of the marginals. Finally, we touch on software specially designed for tail analysis.