Tests of Normality Based on Transformed Empirical Processes

Abstract

The present work introduces four families of tests of normality. The tests in two of them are of the Kolmogorov–Smirnov type, and the tests of the other two are of the Cramer–von Mises type. One family of each type is focused to detect alternatives of skewness and the other one is designed to be specially sensitive to changes in kurtosis. The tests in each family depend on a parameter l: for each integer l, the test statistic involves the computation on the standardized sample points of the Hermite polynomials up to degree l+3. The resulting tests are consistent against all alternative distributions such that at least one of their moments up to order l+3 differ from the corresponding moment of the normal distribution with the same mean and variance. Therefore, a sequence of tests for samples of size n and l=l (n) is consistent against any nonnormal alternative, when limn→∞l(n)=∞. The performance of the proposed tests compares favorably with Shapiro–Wilk and Anderson–Darling omnibus tests, LaRiccia's focused tests, and Kallenberg and Ledwina data driven smooth tests, and the statistics can be easily computed. Hints for their computation are provided.