Model selection and estimation of a component in additive regression

الأصل · EN

Let Y∈ⁿ be a random vector with mean s and covariance matrix σ²PₙPₙ 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∈}, we consider, for any m∈, the least-squares estimator sₘ of s in Sₘ. Considering a penalty function that is not linear in the dimensions of the Sₘ's, we select some m∈ in order to get an estimator sm with a quadratic risk as close as possible to the minimal one among the risks of the sₘ's. Non-asymptotic oracle-type inequalities and minimax convergence rates are proved for sm. 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.