Sample Size Determination for Skewed and Heavy-tailed Distributions

In this article, we propose methods to determine sufficient 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 degree freedom at least three. Motivated by the observed results, we propose a mathematical formula for determining enough sample sizes. The formula is valid for distributions with at least the fourth moment. Then, we propose an algorithm to apply this formula for a data set from general distributions. For distributions with infinite/undefined skew-ness/kurtosis, such as some heavy-tailed distributions, we could use Monte-Carlo simulation method to determine sample size. As an example, we propose an empirical method to determine the sample sizes for Pareto distributions. Both the algorithm and the empirical method are tested on simulated data.