Open Access
December 2018 Optimal adaptive estimation of linear functionals under sparsity
Olivier Collier, Laëtitia Comminges, Alexandre B. Tsybakov, Nicolas Verzelen
Ann. Statist. 46(6A): 3130-3150 (December 2018). DOI: 10.1214/17-AOS1653


We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector $\theta \in\mathbb{R}^{d}$ belongs to a class of $s$-sparse vectors with unknown $s$. We suggest an adaptive estimator achieving a nonasymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when $s$ is unknown. Furthermore, we address the issue of simultaneous adaptation to $s$ and to the variance $\sigma^{2}$ of the noise. We suggest an estimator that achieves the optimal adaptive rate when both $s$ and $\sigma^{2}$ are unknown.


Download Citation

Olivier Collier. Laëtitia Comminges. Alexandre B. Tsybakov. Nicolas Verzelen. "Optimal adaptive estimation of linear functionals under sparsity." Ann. Statist. 46 (6A) 3130 - 3150, December 2018.


Received: 1 November 2016; Revised: 1 October 2017; Published: December 2018
First available in Project Euclid: 7 September 2018

zbMATH: 06968611
MathSciNet: MR3851767
Digital Object Identifier: 10.1214/17-AOS1653

Primary: 62G05 , 62J05

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

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 6A • December 2018
Back to Top