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June 2017 Minimax estimation of linear and quadratic functionals on sparsity classes
Olivier Collier, Laëtitia Comminges, Alexandre B. Tsybakov
Ann. Statist. 45(3): 923-958 (June 2017). DOI: 10.1214/15-AOS1432

Abstract

For the Gaussian sequence model, we obtain nonasymptotic minimax rates of estimation of the linear, quadratic and the $\ell_{2}$-norm functionals on classes of sparse vectors and construct optimal estimators that attain these rates. The main object of interest is the class $B_{0}(s)$ of $s$-sparse vectors $\theta=(\theta_{1},\dots,\theta_{d})$, for which we also provide completely adaptive estimators (independent of $s$ and of the noise variance $\sigma $) having logarithmically slower rates than the minimax ones. Furthermore, we obtain the minimax rates on the $\ell_{q}$-balls $B_{q}(r)=\{\theta\in\mathbb{R}^{d}:\|\theta\|_{q}\le r\}$ where $0<q\le2$, and $\|\theta\|_{q}=(\sum_{i=1}^{d}|\theta_{i}|^{q})^{1/q}$. This analysis shows that there are, in general, three zones in the rates of convergence that we call the sparse zone, the dense zone and the degenerate zone, while a fourth zone appears for estimation of the quadratic functional. We show that, as opposed to estimation of $\theta$, the correct logarithmic terms in the optimal rates for the sparse zone scale as $\log(d/s^{2})$ and not as $\log(d/s)$. For the class $B_{0}(s)$, the rates of estimation of the linear functional and of the $\ell_{2}$-norm have a simple elbow at $s=\sqrt{d}$ (boundary between the sparse and the dense zones) and exhibit similar performances, whereas the estimation of the quadratic functional $Q(\theta)$ reveals more complex effects: the minimax risk on $B_{0}(s)$ is infinite and the sparseness assumption needs to be combined with a bound on the $\ell_{2}$-norm. Finally, we apply our results on estimation of the $\ell_{2}$-norm to the problem of testing against sparse alternatives. In particular, we obtain a nonasymptotic analog of the Ingster–Donoho–Jin theory revealing some effects that were not captured by the previous asymptotic analysis.

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Olivier Collier. Laëtitia Comminges. Alexandre B. Tsybakov. "Minimax estimation of linear and quadratic functionals on sparsity classes." Ann. Statist. 45 (3) 923 - 958, June 2017. https://doi.org/10.1214/15-AOS1432

Information

Received: 1 February 2015; Revised: 1 October 2015; Published: June 2017
First available in Project Euclid: 13 June 2017

zbMATH: 1368.62191
MathSciNet: MR3662444
Digital Object Identifier: 10.1214/15-AOS1432

Subjects:
Primary: 62G05 , 62J05

Keywords: linear functional , Nonasymptotic minimax estimation , quadratic functional , Sparsity , thresholding , unknown noise variance

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • June 2017
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