Goodness-of-fit tests based on sup-functionals of weighted empirical processes

A large class of goodness-of-fit test statistics based on sup-functionals of weighted empirical processes is proposed and studied. The weight functions employed are the Erdős–Feller–Kolmogorov–Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Csörgő, S. Csörgő, L. Horváth, and D. Mason on this type of test statistics, we provide the asymptotic null distribution theory for the class of tests and present an algorithm for tabulating the limit distribution functions under the null hypothesis. A new family of nonparametric confidence bands is constructed for the true distribution function and is found to perform very well. The results obtained, involving a new result on the convergence in distribution of the higher criticism statistic, as introduced by D. Donoho and J. Jin, demonstrate the advantage of our approach over a common approach that utilizes a family of regularly varying weight functions. Furthermore, we show that, in various subtle problems of detecting sparse heterogeneous mixtures, the proposed test statistics achieve the detection boundary found by Yu. I. Ingster and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects.