Model selection and estimation of a component in additive regression

Institut de Mathématiques de Toulouse, Équipe de Statistique et Probabilités, Université Paul Sabatier, 31000 Toulouse, France
xavier.gendre@math.univ-toulouse.fr

Received: 1 February 2011
Revised: 2 August 2012

Abstract

Let Y ∈ ℝⁿ be a random vector with mean s and covariance matrix σ²P_n^tP_n where P_n is some known n × n-matrix. We construct a statistical procedure to estimate s as well as under moment condition on Y or Gaussian hypothesis. Both cases are developed for known or unknown σ². Our approach is free from any prior assumption on s and is based on non-asymptotic model selection methods. Given some linear spaces collection {S_m, m ∈ ℳ}, we consider, for any m ∈ ℳ, the least-squares estimator ŝ_m of s in S_m. Considering a penalty function that is not linear in the dimensions of the S_m’s, we select some m̂ ∈ ℳ in order to get an estimator ŝ_m̂ with a quadratic risk as close as possible to the minimal one among the risks of the ŝ_m’s. Non-asymptotic oracle-type inequalities and minimax convergence rates are proved for ŝ_m̂. A special attention is given to the estimation of a non-parametric component in additive models. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.