On estimation of nonsmooth functionals of sparse normal means

We study the problem of estimation of N-gamma (theta) = Sigma(d)(i=1) vertical bar theta(i)vertical bar(gamma) for gamma > 0 and of the l(gamma)-norm of theta for 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 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.