Overparameterized maximum likelihood tests for detection of sparse vectors

Abstract

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.