Estimation in Ill-posed Linear Models with Nuisance Design

Yuri Golubev (1) , Thomas Zimolo (2, 1)

1 I2M - Institut de Mathématiques de Marseille
2 Université de Provence - Aix-Marseille 1

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 ∈ R^p is a nuisance vector with positive components and ξ, ξ' ∈ R^p are standard white Gaussian noises in R^p. 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/b^2k 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.

Mots clés: secondary 62C10; noisy deconvolution; inverse minimax estimation; Van Trees inequality; roughness penalty approach

Domaines: Statistiques [stat]

Yuri Golubev, Thomas Zimolo. Estimation in Ill-posed Linear Models with Nuisance Design. Mathematical Methods of Statistics, 2015, 24 (1), ⟨10.3103/S1066530715010019⟩. ⟨hal-01287459⟩