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.
