An empirical study of the behaviour of the sample kurtosis in samples from symmetric stable distributions

Abstract

Kurtosis is seen as a measure of the discrepancy between the observed data and a Gaussian distribution and is defined when the fourth moment is finite. In this work an empirical study is conducted to investigate the behaviour of the sample estimate of kurtosis with respect to sample size and the tail index when applied to heavy-tailed data where the fourth moment does not exist. The study will focus on samples from the symmetric stable distributions. It was found that the expected value of excess kurtosis divided by the sample size is finite for any value of the tail index and the sample estimate of excess kurtosis increases as a linear function of sample size and it is approximately equal to n(1 − α/2).