Unification of Rare and Weak Detection Models using Moderate Deviations Analysis and Log-Chisquared P-values
Alon Kipnis
TL;DR
This work develops and unifies Rare and Weak Detection models through Rare Moderate Departures (RMD) analyzed via a log-chisquared P-value framework. It shows that under the global null, $-2\log(p_i)$ behaves like Exp$(2)$, while a vanishingly small fraction follow a scaled noncentral chi-squared tail $Q_i^{(n)}\overset{D}{=}(\mu_n(\rho)+\sigma Z)^2$ with $\mu_n(\rho)=\sqrt{2\rho\log(n)}$, enabling precise phase-transition characterizations. The authors derive the optimal testing regime: Higher Criticism and Berk-Jones achieve maximal asymptotic power outside a delineated powerless region defined by $\rho^*(\beta,\sigma)$, while traditional approaches such as Bonferroni, BH-FDR, minimal P-value, and Fisher's method are suboptimal in general. They also provide extensive model instantiations (heteroscedastic normal, Poisson, binomial) and show log-chisquared tails offer a better approximation than log-normal under moderate deviations, with empirical evidence supporting the theory. The framework yields new results on two-sample heteroscedastic models and perturbed binomial experiments, offering a cohesive, theory-driven guide for high-dimensional inference in sparse settings.
Abstract
Rare and Weak models for multiple hypothesis testing assume that only a small proportion of the tested hypotheses concern non-null effects and the individual effects are only moderately large, so they generally do not stand out individually, for example in a Bonferroni analysis. Such models have been studied in quite a few settings, for example in some cases studies focused on an underlying Gaussian means model for the hypotheses being tested; in others, Poisson and Binomial. Such seemingly different models have the following common structure. Summarizing the evidence of individual tests by the negative logarithm of its P-value, the model is asymptotically equivalent to a situation in which most negative log P-values have a standard exponential distribution but a small fraction might have an alternative distribution which is approximately noncentral chisquared on one degree of freedom. We characterize the asymptotic performance of global tests combining asymptotic log-chisquared P-values in terms of the chisquared mixture parameters: the scaling parameter controlling heteroscedasticity, the non-centrality parameter, and the parameter controlling the rarity of individual non-null effects. In a phase space involving the last two parameters, we derive a region where all tests are asymptotically powerless. Outside of this region, the Berk-Jones and the Higher Criticism tests have maximal power. Inference techniques based on the minimal P-value, false-discovery rate controlling, and Fisher's combination test have sub-optimal asymptotic phase diagrams. Our analysis yields the asymptotic power of global testing in various new rare and weak models, including two-sample heteroscedastic normal mixtures and binomial experiments with perturbed probabilities of success.