Adaptive estimation of the sparsity in the Gaussian vector model

Consider the Gaussian vector model with mean value .. We study the twin problems of estimating the number ...0 of nonzero components of . and testing whether ...0 is smaller than some value. For testing, we establish the minimax separation distances for this model and introduce a minimax adaptive test. Extensions to the case of unknown variance are also discussed. Rewriting the estimation of ...0 as a multiple testing problem of all hypotheses {...0.q}, we both derive a new way of assessing the optimality of a sparsity estimator and we exhibit such an optimal procedure. This general approach provides a roadmap for estimating the complexity of the signal in various statistical models.