On estimation of nonsmooth functionals of sparse normal means

TL;DR

This paper investigates the estimation of a specific nonsmooth functional of sparse normal means, providing minimax risk bounds and proposing estimators that achieve optimal rates.

Contribution

It establishes non-asymptotic minimax risk bounds for estimating the functional and introduces estimators that attain these bounds in the sparse normal means model.

Findings

Derived non-asymptotic minimax risk bounds for the problem

Proposed estimators that achieve the minimax rate

Applicable to the class of s-sparse vectors

Abstract

We study the problem of estimation of the value N_gamma(\theta) = sum(i=1)^d |\theta_i|^gamma for 0 < gamma <= 1 based on the observations y_i = \theta_i + \epsilon\xi_i, i = 1,...,d, where \theta = (\theta_1,...,\theta_d) are unknown parameters, \epsilon>0 is known, and \xi_i are i.i.d. standard normal random variables. We prove that the non-asymptotic minimax risk on the class B_0(s) of s-sparse vectors and we propose estimators achieving the minimax rate.