Test for Normality Using a Characterization

Summary

Let X_i, i=1,…n, be independently and identically distributed. It is well known that the independence of the sample mean and the sample variance characterizes normality; that is, this independence holds if and only if the distribution is normal. Thus, this independence condition summarizes all the properties of the normal. A nonparametric test based on this characterization is given. Power of the test against many alternatives is computed. As should be clear from Table 5, there are a very large number of competing tests, each one with its own merits and demerits. For tests of composite hypothesis of the type tackled here, it is unlikely that an all-round optimal test will ever be achieved. A reason for adding one more test to that list is that the test presented here is based on a property not used before.