Higher criticism for detecting sparse heterogeneous mixtures
Abstract
Higher criticism, or secondlevel significance testing, is a multiplecomparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested comparing the fraction of observed significances at a given \alphalevel to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming a zscore; the resulting zscore tests the significance of the body of significance tests. We consider a generalization, where we maximize this zscore over a range of significance levels 0<\alpha\leq\alpha_0. We are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem: testing whether n normal means are all zero versus the alternative that a small fraction is nonzero. The subtlety of this ``sparse normal means'' testing problem can be seen from work of Ingster and Jin, who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the pvalues either for the bulk of the tests or for the few most highly significant tests. In this range, when the amplitude of nonzero means is calibrated with the fraction of nonzero means, the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2004
 arXiv:
 arXiv:math/0410072
 Bibcode:
 2004math.....10072D
 Keywords:

 Mathematics  Statistics;
 62G10 (Primary) 62G32;
 62G20. (Secondary)
 EPrint:
 Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/009053604000000265