  Consider a multiple hypothesis testing setting involving rare/weak features:
only few features, out of possibly many, deviate from their null hypothesis
behavior. Summarizing the significance of each feature by a P-value, we
construct a global test against the null using the Higher Criticism (HC)
statistics of these P-values. We calibrate the rare/weak model using parameters
controlling the asymptotic behavior of these P-values in their near-zero
"tail". We derive a region in the parameter space where the HC test is
asymptotically powerless. Our derivation involves very different tools than
previously used to show the powerlessness of HC, relying on properties of the
empirical processes underlying HC. In particular, our result applies to
situations where HC is not asymptotically optimal, or when the asymptotically
detectable region of the parameter space is unknown.
