Optimal adaptive estimation of linear functionals under sparsity

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector theta is an element of 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.