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.
"Empirical-likelihood-based confidence interval for the mean with a heavy-tailed distribution." Ann. Statist. 32 (3) 1192 - 1214, June 2004. https://doi.org/10.1214/009053604000000328