## Electronic Journal of Statistics

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

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

Amit Moscovich, Boaz Nadler, and Clifford Spiegelman

#### Abstract

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

**Source**

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

**Dates**

Received: February 2016

First available in Project Euclid: 2 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1472829397

**Digital Object Identifier**

doi:10.1214/16-EJS1172

**Mathematical Reviews number (MathSciNet)**

MR3544289

**Zentralblatt MATH identifier**

1346.62092

**Subjects**

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties

Secondary: 62-04: Explicit machine computation and programs (not the theory of computation or programming)

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.ejs/1472829397

#### Supplemental materials

- Supplementary to “On the exact Berk-Jones statistics and their p-value calculation”. Digital Object Identifier: doi:10.1214/16-EJS1172SUPP