On uniform consistency of Neyman’s type nonparametric tests

Abstract

The goodness-of-fit problem is explored, when the test statistic is a linear combination of squared Fourier coefficients’ estimates coming from the Fourier decomposition of a probability density. Common examples of such statistics include Neyman’s test statistics and test statistics, generated by L2-norms of kernel estimators. We prove the asymptotic normality of the test statistic for both the null and alternative hypothesis. Using these results we deduce conditions of uniform consistency for nonparametric sets of alternatives, which are defined in terms of distribution or density functions. Results on uniform consistency, related to the distribution functions, can be seen as a statement showing to what extent the distance method, based on a given test statistic, makes the hypothesis and alternatives distinguishable. In this case, the deduced conditions of uniform consistency are close to necessary. For sequences of alternatives — defined in terms of density functions — approaching the hypothesis in L2-metric, we find necessary and sufficient conditions for their consistency. This result is obtained in terms of the concept of maxisets, the description of which for given test statistics is found in this publication.