On estimation of nonsmooth functionals of sparse normal means

05/28/2018

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by   Olivier Collier, et al.

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We study the problem of estimation of the value N_gamma(θ) = sum(i=1)^d |θ_i|^gamma for 0 < gamma <= 1 based on the observations y_i = θ_i + ϵξ_i, i = 1,...,d, where θ = (θ_1,...,θ_d) are unknown parameters, ϵ>0 is known, and ξ_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.