%0 Journal Article
%T Spectral cut-off regularizations for ill-posed linear models
%+ Moscow Institute of Physics and Technology [Moscow] (MIPT)
%+ Institut de Mathématiques de Marseille (I2M)
%A Chernousova, E
%A Golubev, Yu
%< avec comité de lecture
%@ 1066-5307
%J Mathematical Methods of Statistics
%I Springer
%8 2014
%D 2014
%R 10.3103/S1066530714020033
%K iIll-posed linear model
%K spectral cut-off regularization
%K data-driven cut-off frequency
%K oracle inequality
%K minimax risk
%K sec-ondary 62C10
%Z 2010 Mathematics Subject Classification: Primary 62C99, secondary 62C10, 62C20, 62J05
%Z Statistics [stat]Journal articles
%X This paper deals with recovering an unknown vector β from the noisy data Y = Xβ + σξ, where X is a known n × p-matrix with n ≥ p and ξ is a standard white Gaussian noise. In order to estimate β, a spectral cutoff estimate β(m,Y) with a data-driven cutoff frequency m(Y) is used. The cutoff frequency is selected as a minimizer of the unbiased risk estimate of the mean square prediction error, i.e. m(Y) = arg min_{m}\| Y − X β (m, Y)\|^2 + 2σ^2 m. Assuming that β belongs to an ellipsoid W, we derive upper bounds for the maximal risk sup_{β∈W}E \|β[m(Y), Y] − β\|^2 and show that β[m(Y), Y] is a rate optimal minimax estimate over W.
%G English
%2 https://hal.science/hal-01292417/document
%2 https://hal.science/hal-01292417/file/spectalcutoff_mms.pdf
%L hal-01292417
%U https://hal.science/hal-01292417
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ INSMI
%~ I2M
%~ I2M-2014-