Optimal adaptive estimation of linear functionals under sparsity

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.