On estimation of nonsmooth functionals of sparse normal means

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.