Overparameterized maximum likelihood tests for detection of sparse vectors

We address the problem of detecting a sparse high-dimensional vector against white Gaussian noise. An unknown vector is assumed to have only p nonzero components, whose positions and sizes are unknown, the number p being on one hand large but on the other hand small as compared to the dimension. The maximum likelihood (ML) test in this problem has a simple form and, certainly, depends of $p$. We study statistical properties of overparametrized ML tests, i.e., those constructed based on the assumption that the number of nonzero components of the vector is $q (q>p)$ in a situation where the vector actually has only p nonzero components. We show that in some cases overparametrized tests can be better than standard ML tests.