Hypothesis Test of a Truncated Sample Mean for the Extremely Heavy-Tailed Distributions

TL;DR

This paper develops hypothesis tests for extremely heavy-tailed distributions with infinite mean or variance using truncated sample means, providing conditions for their asymptotic distributions and illustrating with simulations.

Contribution

It introduces necessary and sufficient conditions for the asymptotic behavior of truncated test statistics in heavy-tailed distributions, advancing statistical inference methods.

Findings

Asymptotic normality under certain conditions

Convergence to stable distributions or negative infinity

Simulation confirms theoretical results

Abstract

This article deals with the hypothesis test for the extremely heavy-tailed distributions with infinite mean or variance by using a truncated sample mean. We obtain three necessary and sufficient conditions under which the asymptotic distribution of the truncated test statistics converges to normal, neither normal nor stable or converges to $-\infty$ or the combination of stable distributions, respectively. The numerical simulation illustrates an application of the theoretical results above in the hypothesis testing.