A Test for Comparing Tail Indices for Heavy-Tailed Distributions via Empirical Likelihood

Abstract

In this work, the problem of testing whether different (⩾2) independent samples, with (possibly) different heavy-tailed distributions, share the same extreme value index, is addressed. The test statistic proposed is inspired by the empirical likelihood methodology and consists in an ANOVA-like confrontation of Hill estimators. Asymptotic validity of this simple procedure is proved and efficiency, in terms of empirical type I error and power, is investigated through simulations under a variety of situations. Surprisingly, this topic had hardly been addressed before, and only in the two-sample case, though it can prove useful in applications.

Keywords:

• Extreme values
• Tail index
• Comparison test
• Hill estimator
• Empirical likelihood.

Mathematics Subject Classification:

• Primary 62G32
• Secondary 62G10.