Composite hypothesis tests for sparse parameters

In an estimation framework, the suspected sparsity of an unknown vector of deterministic parameters is classically accounted for through thresholding functions, some of which being related to Maximum A Posteriori estimates with specific priors. In a detection framework, statistical tests applied to sparse vectors are classically designed so as to limit the focus of the test to a few active components, leading to test statistics of thresholded data. We propose a study of the connections between these two problems. The detection tests we consider are the Generalized Likelihood Ratio Test (GLRT), the Bayes Factor, the Posterior Density Ratio and a LRT using a Maximum A Posteriori estimate with Generalized Gaussian priors. In the case of scalar parameter first, we derive sufficient conditions that any detection test must obey to be equivalent to a GLRT, and prove that the tests above verify them. In the vector case, the LRT with a MAP estimate and the PDR outperform the GLRT and the BF thanks to the thresholding effect of the MAP estimation. The connections between thresholding functions used in estimation and the resulting ”thresholded test statistics” are precisely investigated.