Empirical Likelihood based Confidence Regions for first order parameters of a heavy tailed distribution

TL;DR

This paper develops empirical likelihood-based confidence regions for the tail index and scale parameters of heavy-tailed distributions, demonstrating improved performance over traditional Wald-type methods through asymptotic validity.

Contribution

It introduces a novel empirical likelihood approach combined with estimation equations for heavy-tailed distribution parameters, providing more accurate confidence regions.

Findings

Confidence regions outperform Wald-type intervals.

Asymptotic validity of the proposed confidence regions.

Effective profiling for tail index confidence intervals.

Abstract

Let $X_1, \ldots, X_n$ be some i.i.d. observations from a heavy tailed distribution $F$, i.e. such that the common distribution of the excesses over a high threshold $u_n$ can be approximated by a Generalized Pareto Distribution $G_{\gamma,\sigma_n}$ with $\gamma >0$. This work is devoted to the problem of finding confidence regions for the couple $(\gamma,\sigma_n)$ : combining the empirical likelihood methodology with estimation equations (close but not identical to the likelihood equations) introduced by J. Zhang (Australian and New Zealand J. Stat n.49(1), 2007), asymptotically valid confidence regions for $(\gamma,\sigma_n)$ 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.