On the Detection of Sparse Mixtures

This thesis focuses on two topics. In the first topic, 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. In the second topic, we consider the problem of detecting a sparse Poisson mixture. Our results parallel those for the detection of a sparse normal mixture, pioneered by Ingster (1997) and Donoho and Jin (2004), when the Poisson means are larger than logarithmic in the sample size. In particular, a form of higher criticism achieves the detection boundary in the whole sparse regime. When the Poisson means are smaller than logarithmic in the sample size, a different regime arises in which simple multiple testing with Bonferroni correction is enough in the sparse regime. We present some numerical experiments that confirm our theoretical finding