Tests for Normality Based on Entropy Divergences

The normal distribution is among the most useful distributions in statistical applications. Accordingly, testing for normality is of fundamental importance in many fields including biopharmaceutical research. A generally powerful test for normality is the Shapiro-Wilk test, which can be derived based on estimated entropy divergence. Another well-known test for normality based on entropy divergence was proposed by Vasicek (1976) which has inspired the development of many goodness-of-fit tests for other important distributions. Despite extensive research on the subject, there still exists considerable confusion concerning the fundamental characteristics of Vasicek’s test. This article presents a unified derivation of both the Shapiro-Wilk test and Vasicek’s test based on estimated entropy divergence and clarifies some existing confusion. A comparative study of power performance for these two well-known tests for normality is presented with respect to a wide range of alternatives.