Model selection and estimation of a component in additive regression. (28th November 2013) - Gendre, Xavier
Abstract : Let Y  ∈ ℝ n be a random vector with mean s and covariance matrix σ 2 P n t P 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 σ 2 . 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.