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 θ ∈ Rd 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 σ2 of the noise. We suggest an estimator that achieves the optimal adaptive rate when both s and σ2 are unknown.