  We adapt Higher Criticism (HC) to the comparison of two frequency tables
which may -- or may not -- exhibit moderate differences between the tables in
some unknown, relatively small subset out of a large number of categories. Our
analysis of the power of the proposed HC test quantifies the rarity and size of
assumed differences and applies moderate deviations-analysis to determine the
asymptotic powerfulness/powerlessness of our proposed HC procedure.
  Our analysis considers the null hypothesis of no difference in underlying
generative model against a rare/weak perturbation alternative, in which the
frequencies of $N^{1-\beta}$ out of the $N$ categories are perturbed by $r(\log
N)/2n$ in the Hellinger distance; here $n$ is the size of each sample. Our
proposed Higher Criticism (HC) test for this setting uses P-values obtained
from $N$ exact binomial tests. We characterize the asymptotic performance of
the HC-based test in terms of the rarity parameter $\beta$ and the perturbation
intensity parameter $r$. Specifically, we derive a region in the
$(\beta,r)$-plane where the test asymptotically has maximal power, while having
asymptotically no power outside this region. Our analysis distinguishes between
cases in which the counts in both tables are low, versus cases in which counts
are high, corresponding to the cases of sparse and dense frequency tables. The
phase transition curve of HC in the high-counts regime matches formally the
curve delivered by HC in a two-sample normal means model.
