  Detection of sparse signals arises in a wide range of modern scientific
studies. The focus so far has been mainly on Gaussian mixture models. In this
paper, we consider the detection problem under a general sparse mixture model
and obtain an explicit expression for the detection boundary. It is shown that
the fundamental limits of detection is governed by the behavior of the
log-likelihood ratio evaluated at an appropriate quantile of the null
distribution. We also establish the adaptive optimality of the higher criticism
procedure across all sparse mixtures satisfying certain mild regularity
conditions. In particular, the general results obtained in this paper recover
and extend in a unified manner the previously known results on sparse detection
far beyond the conventional Gaussian model and other exponential families.
