  Consider a multiple hypothesis testing setting in which only a small
proportion of the measured features contain non-null effects. Under typical
conditions, the log of the P-value associated with each feature is
approximately a sparse mixture of chi-squared distributions, one of which is
scaled and non-central. We characterize the asymptotic performance of global
tests combining these P-values in terms of the chisquared mixture parameters:
the scaling parameters controlling heteroscedasticity, the non-centrality
parameter describing the effect size, and the parameter controlling the rarity
of the non-null features. Specifically, 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 of these
P-values have maximal power. Inference techniques based on the minimal P-value,
false-discovery rate controlling, and Fisher's combination test have
sub-optimal asymptotic performance. We provide various examples for recently
studied signal detection models that fall under our setting as well as several
new ones. Our perturbed log-chisquared P-values formulation seamlessly
generalizes these models to their two-sample variant and heteroscedastic
situations. The log-chisquared approximation for P-values under the alternative
hypothesis is different from Bahadur's classical log-normal approximation. The
latter turns out to be unsuitable for analyzing optimal detection in rare/weak
feature models.
