The Annals of Statistics

Optimal adaptive estimation of linear functionals under sparsity

Olivier Collier, Laëtitia Comminges, Alexandre B. Tsybakov, and Nicolas Verzelen

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Abstract

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.

Article information

Source
Ann. Statist., Volume 46, Number 6A (2018), 3130-3150.

Dates
Received: November 2016
Revised: October 2017
First available in Project Euclid: 7 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1536307245

Digital Object Identifier
doi:10.1214/17-AOS1653

Mathematical Reviews number (MathSciNet)
MR3851767

Zentralblatt MATH identifier
06968611

Subjects
Primary: 62J05: Linear regression 62G05: Estimation

Keywords
Nonasymptotic minimax estimation adaptive estimation linear functional sparsity unknown noise variance

Citation

Collier, Olivier; Comminges, Laëtitia; Tsybakov, Alexandre B.; Verzelen, Nicolas. Optimal adaptive estimation of linear functionals under sparsity. Ann. Statist. 46 (2018), no. 6A, 3130--3150. doi:10.1214/17-AOS1653. https://projecteuclid.org/euclid.aos/1536307245


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