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

Let X1,…,Xn be some i.i.d. observations from a heavy-tailed distribution F, i.e. the common distribution of the excesses over a high threshold un can be approximated by a generalized Pareto distribution Gγ,σn with γ>0. This paper deals with the problem of finding confidence regions for the couple (γ,σn): 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,σn) 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.