%0 Journal Article
%T Estimation in Ill-posed Linear Models with Nuisance Design
%+ Institut de Mathématiques de Marseille (I2M)
%+ Université de Provence - Aix-Marseille 1
%A Golubev, Yuri
%A Zimolo, Thomas
%< avec comité de lecture
%@ 1066-5307
%J Mathematical Methods of Statistics
%I Springer
%V 24
%N 1
%8 2015
%D 2015
%R 10.3103/S1066530715010019
%K secondary 62C10
%K noisy deconvolution
%K inverse minimax estimation
%K Van Trees inequality
%K roughness penalty approach
%Z Primary 62C99, secondary 62C10, 62C20, 62J05.
%Z Statistics [stat]Journal articles
%X 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.
%G English
%2 https://hal.science/hal-01287459/document
%2 https://hal.science/hal-01287459/file/golubev-zimolo.pdf
%2 https://hal.science/hal-01287459/file/Fig-1.pdf
%2 https://hal.science/hal-01287459/file/Fig-2.pdf
%2 https://hal.science/hal-01287459/file/Fig-3.pdf
%L hal-01287459
%U https://hal.science/hal-01287459
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-