Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 62G05 Estimation
- 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
- 62J05 Linear regression

**Résumé:** This paper is mainly devoted to a precise analysis of what kind of penalties should be used in
order to perform model selection via the minimization of a penalized least-squares type
criterion within some general Gaussian framework. As compared to our previous paper on
this topic (Birg\'e and Massart, 2001), more elaborate forms of the penalties are given which
are shown to be, in some sense, optimal. We also provide risk bounds with explicit absolute
constants and an asymptotic evaluation of the risk which generalizes the one of Shibata (1981)
to our new penalties. Some applications to the estimation of change points for a signal in
Gaussian noise are also developed. We finally present a practical strategy, based on sharp
lower bounds for the penalty function, to design the penalty from the data when the amount of
noise is unknown.

**Mots Clés:** *Gaussian linear regression ; variable selection ; model selection ; Mallows' $C_p$ ; penalized least-squares*

**Date:** 2001-04-02

**Prépublication numéro:** *PMA-647*