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On the exact Berk-Jones statistics and their \(p\)-value calculation. (English) Zbl 1346.62092

Summary: Continuous goodness-of-fit testing is a classical problem in statistics. Despite having low power for detecting deviations at the tail of a distribution, the most popular test is based on the Kolmogorov-Smirnov statistic. While similar variance-weighted statistics such as Anderson-Darling and the Higher Criticism statistic give more weight to tail deviations, as shown in various works, they still mishandle the extreme tails.
As a viable alternative, in this paper we study some of the statistical properties of the exact \(M_{n}\) statistics of Berk and Jones. In particular we show that they are consistent and asymptotically optimal for detecting a wide range of rare-weak mixture models. Additionally, we present a new computationally efficient method to calculate \(p\)-values for any supremum-based one-sided statistic, including the one-sided \(M_{n}^{+},M_{n}^{-}\) and \(R_{n}^{+},R_{n}^{-}\) statistics of Berk and Jones and the Higher Criticism statistic. Finally, we show that \(M_{n}\) compares favorably to related statistics in several finite-sample simulations.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62-04 Software, source code, etc. for problems pertaining to statistics
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