Adaptive estimation of the sparsity in the Gaussian vector model
Alexandra Carpentier, Nicolas Verzelen
The Annals of Statistics · 2019 · Volume 47 · Issue 1
Abstract
Consider the Gaussian vector model with mean value $\theta$. We study the twin problems of estimating the number $\Vert \theta \Vert_{0}$ of nonzero components of $\theta$ and testing whether $\Vert \theta \Vert_{0}$ 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 $\Vert \theta \Vert_{0}$ as a multiple testing problem of all hypotheses $\{\Vert \theta \Vert_{0}\leq q\}$, 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.
Keywords
62B10, 62C20, 62G10, composite-composite testing problems, minimax separation distance in testing problems, Sparsity estimation and testing
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