Rare and Weak Detection Models under Moderate Deviations Analysis and Log-Chisquared P-values

Rare/Weak models for multiple hypothesis testing assume that only a small proportion of the tested hypotheses concern non-null effects and the individual signals are only moderately large, so that they generally do not stand out individually above the noise level. Such models have been studied in quite a few settings, for example in some cases studies focused on underlying Gaussian means model for the hypotheses being tested; in some others, Poisson. It seems not to have been noticed before that such seemingly different models have asymptotically the following common structure: Summarizing the evidence each test provides by the negative logarithm of its P-value, previous rare/weak model settings are asymptotically equivalent to detection where most negative log P-values have a standard exponential distribution but a small fraction of the P-values might have an alternative distribution which is moderately larger; we do not know which individual tests those might be, or even if there are any such. Moreover, the alternative distribution is approximately noncentral chisquared on one degree of freedom. We characterize the asymptotic performance of several global tests combining these P-values using a phase diagram analysis involving the log-chisquared mixture parameters. Interestingly, the log-chisquared approximation for P-values we use here is different from a classical analysis proposing the log-normal approximation which would be unsuitable for understanding rare/weak multiple testing models.