Modification of some goodness-of-fit statistics to yield asymptotically normal null distributions

Abstract

SUMMARY The Cramér-von Mises class of test statistics for the one-sample goodness-of-fit problem is based on the weighted L 2 -distance of the distribution from its empirical counterpart. It is well known that the root π standardized norm has a degenerate limiting distribution under the null hypothesis, and the π standardized norm has a nondegenerate limiting distribution that is not normal and depends on identifying the principal components for every weight function. Usually this limiting distribution is an infinite weighted sum of Chi-squared distributions. In the current investigation, we offer an alternative estimate of the L 2 -norm that is asymptotically normal under both the null and the alternative distributions. Similar modification is indicated for the Watson (1961) statistic used for testing a distribution defined on a circle.