Empirical-likelihood-based confidence interval for the mean with a heavy-tailed distribution

The Annals of Statistics · 2004 · Volume 32 · Issue 3

Abstract

Empirical-likelihood-based confidence intervals for a mean were introduced by Owen [Biometrika 75 (1988) 237–249], where at least a finite second moment is required. This excludes some important distributions, for example, those in the domain of attraction of a stable law with index between 1 and 2. In this article we use a method similar to Qin and Wong [Scand. J. Statist. 23 (1996) 209–219] to derive an empirical-likelihood-based confidence interval for the mean when the underlying distribution has heavy tails. Our method can easily be extended to obtain a confidence interval for any order of moment of a heavy-tailed distribution.

Keywords

62G15, 62G30, Empirical likelihood method, heavy tail, Hill estimator, Normal approximation, Stable law, tail index

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