The Annals of Statistics

Adaptive estimation of the sparsity in the Gaussian vector model

Alexandra Carpentier and Nicolas Verzelen

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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.

Article information

Ann. Statist., Volume 47, Number 1 (2019), 93-126.

Received: March 2017
Revised: September 2017
First available in Project Euclid: 30 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62G10: Hypothesis testing
Secondary: 62B10: Information-theoretic topics [See also 94A17]

Sparsity estimation and testing composite-composite testing problems minimax separation distance in testing problems


Carpentier, Alexandra; Verzelen, Nicolas. Adaptive estimation of the sparsity in the Gaussian vector model. Ann. Statist. 47 (2019), no. 1, 93--126. doi:10.1214/17-AOS1680.

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Supplemental materials

  • Supplement to “Adaptive estimation of the sparsity in the Gaussian vector model”. Proofs of the results.