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A note on goodness-of-fit statistics with asymptotically normal distributions

A generalization of the Cramer-vonMises L2 distance is proposed. It gives rise to a class of goodness-of-fit statistics that is difficult to analyze using traditional techniques based on empirical distributions but can easily be modified to yield null and non null limiting normal distributions. The family index may be used to maximize the power of the test for a specific alternative hypothesis. The procedure presented here is shown to work for Watson's modification for circular data and also when testing symmetry about the zero. The problem of testing two-samples is also presented. All procedures presented here are distributions-free and can be used equally for univariate or multivariate data.