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Model selection and estimationof a component in additive regression

Published online by Cambridge University Press:  28 November 2013

Xavier Gendre
Xavier Gendre*
Affiliation:
Institut de Mathématiques de Toulouse, Équipe de Statistique et Probabilités, Université Paul Sabatier, 31000 Toulouse, France. [email protected]
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Abstract

Let Y ∈ ℝn be a random vector with means and covariance matrix σ2PntPn wherePn is some knownn × n-matrix. We construct a statistical procedure toestimate s as well as under moment condition on Y orGaussian hypothesis. Both cases are developed for known or unknownσ2. Our approach is free from any prior assumption ons and is based on non-asymptotic model selection methods. Given somelinear spaces collection {Sm, m ∈ ℳ}, we consider, for any m ∈ ℳ, the least-squaresestimator ŝm of s inSm. Considering a penalty function that isnot linear in the dimensions of the Sm’s, weselect some m̂ ∈ ℳ in order to get an estimatorŝm̂ with a quadratic risk as close aspossible to the minimal one among the risks of theŝm’s. Non-asymptotic oracle-typeinequalities and minimax convergence rates are proved forŝm̂. A special attention is given to theestimation of a non-parametric component in additive models. Finally, we carry out asimulation study in order to illustrate the performances of our estimators inpractice.

Keywords

Model selectionnonparametric regressionpenalized criterionoracle inequalitycorrelated dataadditive regressionminimax rate
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 77 - 116
Copyright
© EDP Sciences, SMAI, 2013

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