# Chapter 35A Test for Normality Based on Sample Entropy

*J. Roy. Statist. Soc*., Series B, 38 (1) (1976), 54–59.

## Abstract

This chapter introduces a test of the composite hypothesis of normality. The test is based on the property of the normal distribution that its entropy exceeds that of any other distribution with a density that has the same variance. The test statistic is based on a class of estimators of entropy constructed here. The test is shown to be a consistent test of the null hypothesis for all alternatives without a singular continuous part. The power of the test is estimated against several alternatives. It is observed that the test compares favorably with other tests for normality.

## Entropy Estimation

The entropy of a distribution *F* with a density function *f* is defined as

Let , be a sample from the distribution *F*. Express (1) in the form

An estimate of (2) can be constructed by replacing the distribution function *F* by the empirical distribution function , and using a difference operator in place of the differential operator. The derivative of is then estimated by ...