Model selection and estimation of a component in additive regression
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 77-116.

Let Y ∈ ℝn be a random vector with mean s and covariance matrix σ2PntPn where Pn 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 {Sm, m ∈ ℳ}, we consider, for any m ∈ ℳ, the least-squares estimator ŝm of s in Sm. Considering a penalty function that is not linear in the dimensions of the Sm's, we select some m̂ ∈ ℳ in order to get an estimator ŝm̂ with a quadratic risk as close as possible to the minimal one among the risks of the ŝm's. Non-asymptotic oracle-type inequalities and minimax convergence rates are proved for ŝm̂. 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.

DOI : 10.1051/ps/2012028
Classification : 62G08
Mots-clés : model selection, nonparametric regression, penalized criterion, oracle inequality, correlated data, additive regression, minimax rate 
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     author = {Gendre, Xavier},
     title = {Model selection and estimation of a component in additive regression},
     journal = {ESAIM: Probability and Statistics},
     pages = {77--116},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2012028},
     mrnumber = {3143734},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2012028/}
}
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Gendre, Xavier. Model selection and estimation of a component in additive regression. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 77-116. doi : 10.1051/ps/2012028. http://archive.numdam.org/articles/10.1051/ps/2012028/

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