Estimation in ill-posed linear models with nuisance design

Abstract

The paper deals with recovering an unknown vector θ ∈ ℝp in two simple linear models: in the first one we observe y = b · θ + ∈ξ and z = b + σξ′, whereas in the second one we have at our disposal y′ = b2 · θ + ∈b · ξ and z = b + σξ′. Here b ∈ ℝp is a nuisance vector with positive components and ξ, ξ′ ∈ ℝp are standard white Gaussian noises in ℝp. It is assumed that p is large and components bk of b are small for large k. In order to get good statistical estimates of θ in this situation, we propose to combine minimax estimates of 1/bk and 1/bk2 with regularization techniques based on the roughness penalty approach. We provide new non-asymptotic upper bounds for the mean square risks of the estimates related to this method.