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′ = b 2 · 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 b k 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/b k and 1/b 2 k 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.
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Golubev, Y., Zimolo, T. Estimation in ill-posed linear models with nuisance design. Math. Meth. Stat. 24, 1–15 (2015). https://doi.org/10.3103/S1066530715010019
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DOI: https://doi.org/10.3103/S1066530715010019