The Annals of Statistics

Minimax estimation of linear and quadratic functionals on sparsity classes

Olivier Collier, Laëtitia Comminges, and Alexandre B. Tsybakov

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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.

Article information

Ann. Statist., Volume 45, Number 3 (2017), 923-958.

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

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Zentralblatt MATH identifier

Primary: 62J05: Linear regression 62G05: Estimation

Nonasymptotic minimax estimation linear functional quadratic functional sparsity unknown noise variance thresholding


Collier, Olivier; Comminges, Laëtitia; Tsybakov, Alexandre B. Minimax estimation of linear and quadratic functionals on sparsity classes. Ann. Statist. 45 (2017), no. 3, 923--958. doi:10.1214/15-AOS1432.

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