Sample Size Determination for Skewed and Heavy-Tailed Distributions

Abstract

In this article, we propose methods to determine adequate sample sizes for applying the classical central limit theorem to skewed and heavy-tailed distributions. In doing so, we review the properties of an α−stable distribution and its domain of attraction. Then, we apply the general Edgeworth expansion for regularly varying distributions to t−distributions with at least three degrees of freedom. Motivated by the observed results, we propose a mathematical formula for determining adequate sample size. The formula is valid for distributions with at least fourth moments. Then, we propose an algorithm to apply this formula to a data set from general distributions. For distributions with infinite/undefined skewness/kurtosis, such as some heavy-tailed distributions, we could use the Monte-Carlo simulation method to determine adequate sample size. As an example, we propose an empirical method to determine the sample size for Pareto distributions. Both the algorithm and the empirical method are tested on simulated data.

Department: Mathematical Sciences 

Faculty Mentor: Dr. Rui Hu