Finance, Economics, and Mathematics

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

1

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

2

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 ...