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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.
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 ...