On estimation of nonsmooth functionals of sparse normal means

Abstract

We study the problem of estimation of $N_{\gamma }(\theta )=\sum_{i=1}^{d}|\theta _{i}|^{\gamma }$ for $\gamma >0$ and of the $\ell _{\gamma }$-norm of $\theta $ for $\gamma \ge 1$ based on the observations $y_{i}=\theta _{i}+\varepsilon \xi _{i}$, $i=1,\ldots,d$, where $\theta =(\theta _{1},\dots ,\theta _{d})$ are unknown parameters, $\varepsilon >0$ is known, and $\xi _{i}$ are i.i.d. standard normal random variables. We find the non-asymptotic minimax rate for estimation of these functionals on the class of $s$-sparse vectors $\theta $ and we propose estimators achieving this rate.