Selecting the Smoothing Parameter in Goodness of Fit Testing

A crucial ingredient of the Neyman (1937) “smooth type” tests for goodness-of-fit is the selection of the smoothing parameter, which is influenced dramatically by one’s notion of optimality. This is illustrated with the test based on the L1 difference between the histogram estimator and its expectation under the null hypothesis. We determine asymptotically the minimum power β(n,m, δ) as a function of the sample size n, the number of cells m, and δ, the L1 distance between the expectation of the histogram estimator and its expectation under the null. Necessary and (almost) sufficient conditions are identified under which the minimum power tends to 1. We conclude that the optimal number of cells m in the histogram can vary from m = 2 to m n, depending on the optimality criterion.