# Model selection and estimation of a component in additive regression

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1 IMT - Institut de Mathématiques de Toulouse UMR5219

Abstract : Let $Y\in\R^n$ be a random vector with mean $s$ and covariance matrix $\sigma^2P n\tra{P n}$ where $P n$ is some known $n\times n$-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 $\sigma^2$. Our approach is free from any prior assumption on $s$ and is based on non-asymptotic model selection methods. Given some linear spaces collection $\{S m,\ m\in\M\}$, we consider, for any $m\in\M$, the least-squares estimator $\hat{s} m$ of $s$ in $S m$. Considering a penalty function that is not linear in the dimensions of the $S m$-s, we select some $\hat{m}\in\M$ in order to get an estimator $\hat{s} {\hat{m}}$ with a quadratic risk as close as possible to the minimal one among the risks of the $\hat{s} m$-s. Non-asymptotic oracle-type inequalities and minimax convergence rates are proved for $\hat{s} {\hat{m}}$. 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.

keyword : Model Selection Nonparametric Regression Penalized Criterion Oracle Inequality Correlated Data Additive Regression Minimax Rate

Autor: Xavier Gendre -

Fuente: https://hal.archives-ouvertes.fr/

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