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.