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February 2019 Adaptive estimation of the sparsity in the Gaussian vector model
Alexandra Carpentier, Nicolas Verzelen
Ann. Statist. 47(1): 93-126 (February 2019). DOI: 10.1214/17-AOS1680


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.


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Alexandra Carpentier. Nicolas Verzelen. "Adaptive estimation of the sparsity in the Gaussian vector model." Ann. Statist. 47 (1) 93 - 126, February 2019.


Received: 1 March 2017; Revised: 1 September 2017; Published: February 2019
First available in Project Euclid: 30 November 2018

zbMATH: 07036196
MathSciNet: MR3909928
Digital Object Identifier: 10.1214/17-AOS1680

Primary: 62C20 , 62G10
Secondary: 62B10

Keywords: composite-composite testing problems , minimax separation distance in testing problems , Sparsity estimation and testing

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 1 • February 2019
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