Distribution-free tests for sparse heterogeneous mixtures

Abstract

We consider the problem of detecting sparse heterogeneous mixtures from a nonparametric perspective. Specifically, we assume that the null distribution is symmetric about zero, while the true effects have positive median. We then suggest two new tests for this purpose. The main one is a form of Anderson–Darling test for symmetry and is closely related to the higher criticism. It is shown to achieve the detection boundary for the normal mixture model and, more generally, for asymptotically generalized Gaussian mixture models, in all sparsity regimes. The other test is a form of longest run test and specifically designed for the very sparse situation.