Electronic Journal of Statistics

On the exact Berk-Jones statistics and their $p$-value calculation

Amit Moscovich, Boaz Nadler, and Clifford Spiegelman

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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.

Article information

Electron. J. Statist. Volume 10, Number 2 (2016), 2329-2354.

Received: February 2016
First available in Project Euclid: 2 September 2016

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Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties
Secondary: 62-04: Explicit machine computation and programs (not the theory of computation or programming)

Continuous goodness-of-fit Hypothesis testing p-value computation Rare-weak model


Moscovich, Amit; Nadler, Boaz; Spiegelman, Clifford. On the exact Berk-Jones statistics and their $p$-value calculation. Electron. J. Statist. 10 (2016), no. 2, 2329--2354. doi:10.1214/16-EJS1172. http://projecteuclid.org/euclid.ejs/1472829397.

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