  We consider the problem of detecting sparse heterogeneous mixtures from a
nonparametric perspective, and develop distribution-free tests when all effects
have the same sign. Specifically, we assume that the null distribution is
symmetric about zero, while the true effects have positive median. We evaluate
the precise performance of classical tests for the median (t-test, sign test)
and classical tests for symmetry (signed-rank, Smirnov, total number of runs,
longest run tests) showing that none of them is asymptotically optimal for the
normal mixture model in all sparsity regimes. We then suggest two new tests.
The main one is a form of Higher Criticism, or Anderson-Darling, test for
symmetry. It is shown to be asymptotically optimal for the normal mixture
model, and other generalized Gaussian mixture models, in all sparsity regimes.
Our numerical experiments confirm our theoretical findings.
