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

Let X₁,…,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,σn 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.