Empirical likelihood based confidence regions for first order parameters of heavy-tailed distributions

Abstract

Let X 1,…, X n be some i.i.d. observations from a heavy-tailed distribution F, i.e. the common distribution of the excesses over a high threshold u n can be approximated by a generalized Pareto distribution G γ , σ n with γ > 0 . This paper deals with the problem of finding confidence regions for the couple ( γ , σ n ) : combining the empirical likelihood methodology with estimation equations (close but not identical to the likelihood equations) introduced by Zhang (2007), asymptotically valid confidence regions for ( γ , σ n ) are obtained and proved to perform better than Wald-type confidence regions (especially those derived from the asymptotic normality of the maximum likelihood estimators). By profiling out the scale parameter, confidence intervals for the tail index are also derived.