  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 some
others, Poisson and Binomial. Such seemingly different models have
asymptotically 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 of the P-values
might have an alternative distribution which is approximately noncentral
chisquared on one degree of freedom. This log-chisquared approximation is
different from the log-normal approximation of Bahadur which is unsuitable for
analyzing Rare and Weak multiple testing models.
  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 describing the effect size whenever it exists, and the
parameter controlling the rarity of the 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.
