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Higher criticism for detecting sparse heterogeneous mixtures. (English) Zbl 1092.62051
Summary: Higher criticism, or second-level significance testing, is a multiple-comparisons concept mentioned in passing by J. W. Tukey [The higher criticism. Course Notes Stat. 411, Princeton Univ. (1976)]. 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 a-level to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming a z-score; the resulting z-score tests the significance of the body of significance tests. We consider a generalization, where we maximize this z-score over a range of significance levels $$0<a\leq\infty$$. 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 Y. I. Ingster [Math. Methods. Stat. 6, 47–69 (1997; Zbl 0878.62005)] and J. Jin [Detection boundary for sparse mixtures. Unpubl. manuscript. (2002)], 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 p-values 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.
We show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test would succeed. Since it does not require a specification of the alternative, this shows that higher criticism is in a sense optimally adaptive to unknown sparsity and size of the nonnull effects. While our theoretical work is largely asymptotic, we provide simulations in finite samples and suggest some possible applications. We also show that higher critcism works well over a range of non-Gaussian cases.

##### MSC:
 62G10 Nonparametric hypothesis testing 62J15 Paired and multiple comparisons; multiple testing 62G30 Order statistics; empirical distribution functions 62G20 Asymptotic properties of nonparametric inference 62G32 Statistics of extreme values; tail inference
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