## Institute of Mathematical Statistics Collections

### The average likelihood ratio for large-scale multiple testing and detecting sparse mixtures

Guenther Walther

#### Abstract

Large-scale multiple testing problems require the simultaneous assessment of many p-values. This paper compares several methods to assess the evidence in multiple binomial counts of p-values: the maximum of the binomial counts after standardization (the “higher-criticism statistic”), the maximum of the binomial counts after a log-likelihood ratio transformation (the “Berk–Jones statistic”), and a newly introduced average of the binomial counts after a likelihood ratio transformation. Simulations show that the higher criticism statistic has a superior performance to the Berk–Jones statistic in the case of very sparse alternatives (sparsity coefficient $\beta \gtrapprox 0.75$), while the situation is reversed for $\beta \lessapprox 0.75$. The average likelihood ratio is found to combine the favorable performance of higher criticism in the very sparse case with that of the Berk–Jones statistic in the less sparse case and thus appears to dominate both statistics. Some asymptotic optimality theory is considered but found to set in too slowly to illuminate the above findings, at least for sample sizes up to one million. In contrast, asymptotic approximations to the critical values of the Berk–Jones statistic that have been developed by [In High Dimensional Probability III (2003) 321–332 Birkhäuser] and [ Ann. Statist. 35 (2007) 2018–2053] are found to give surprisingly accurate approximations even for quite small sample sizes.

#### Chapter information

Source
Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., and Maathuis, M. H., eds., From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 317-326

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.imsc/1362751197

Digital Object Identifier
doi:10.1214/12-IMSCOLL923

Mathematical Reviews number (MathSciNet)
MR3202643

Zentralblatt MATH identifier
1356.62095

Rights

#### Citation

Walther, Guenther. The average likelihood ratio for large-scale multiple testing and detecting sparse mixtures. From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, 317--326, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL923. https://projecteuclid.org/euclid.imsc/1362751197

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