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 studied. The weight functions employed are Erdos-Feller-Kolmogorov-Petrovski upper-class functions of a Brownian bridge. The whole class of test statistics is shown to be consistent against a fixed alternative. Based on the result of Csorgo et al. [Ann. Probab. 14 (1986) 31-85] obtained for this type of test statistics, we provide the asymptotic null distribution theory for the class of tests at hand, 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 it is found to perform very well. The results obtained, together with a new result on the convergence in distribution of the higher criticism statistic introduced in Donoho and Jin [Ann. Statist. 32 (2004) 962-994] 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 Ingster [Math. Meth. Statist. 6 (1997) 47-69] and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects.