%0 Journal Article
%T ADAPTIVE ESTIMATION UNDER SINGLE-INDEX CONSTRAINT IN A REGRESSION MODEL
%+ Institut de Mathématiques de Marseille (I2M)
%A Lepski, Oleg
%A Serdyukova, Nora
%< avec comité de lecture
%@ 0090-5364
%J Annals of Statistics
%I Institute of Mathematical Statistics
%V 42
%N 1
%P 1-28.
%8 2014-01-28
%D 2014
%R 10.1214/13-AOS1152
%K Adaptive estimation
%K lower bounds
%K minimax rate
%K nonparametric regression
%K oracle inequalities
%K single-index model
%K structural adaptation
%Z Mathematics [math]Journal articles
%X The problem of adaptive multivariate function estimation in the single-index regression model with random design and weak assumptions on the noise is investigated. A novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothness of the link function by selecting from a family of specific kernel estimators is proposed. We establish a pointwise oracle inequality which, in its turn, is used to judge the quality of estimating the entire function (" global " oracle inequality). Both the results are applied to the problems of pointwise and global adaptive estimation over a collection of Hölder and Nikol'skii functional classes, respectively.
%G English
%2 https://hal.science/hal-01265248/document
%2 https://hal.science/hal-01265248/file/Single-Index-AOS.pdf
%L hal-01265248
%U https://hal.science/hal-01265248
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-