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

TL;DR

This paper introduces a new class of goodness-of-fit tests based on sup-functionals of weighted empirical processes with Erdős-Feller-Kolmogorov-Petrovski weights, providing asymptotic null distribution theory, algorithms, and applications to sparse mixture detection.

Contribution

The paper develops a novel class of goodness-of-fit tests using weighted empirical processes, with theoretical null distribution results and practical algorithms, outperforming traditional methods in sparse mixture detection.

Findings

Asymptotic null distribution theory established for the proposed tests.

Algorithm for tabulating limit distribution functions under the null hypothesis.

Proposed tests achieve optimal detection boundaries in sparse mixture problems.

Abstract

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 Erd\H{o}s-Feller-Kolmogorov-Petrovski upper-class functions of a Brownian bridge. Based on the result of M. Cs\"{o}rg\H{o}, S. Cs\"{o}rg\H{o}, Horv\'{a}th, and Mason obtained for this type of test statistics, we provide the asymptotic null distribution theory for the class of tests in 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 by Donoho and 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 Ingster and, when distinguishing between the null and alternative hypotheses, perform optimally adaptively to unknown sparsity and size of the non-null effects.