Open Access
Translator Disclaimer
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

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.

Citation

Download Citation

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

Information

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

Subjects:
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

JOURNAL ARTICLE
34 PAGES


SHARE
Vol.47 • No. 1 • February 2019
Back to Top