**RISK BOUNDS FOR MODEL SELECTION VIA PENALISATION**

**Auteurs : **Andrew Barron (Yale University), Lucien Birgé
(Université Paris VI), Pascal Massart (Université Paris
Sud)

**Mots Clés :** Penalization, Model selection, Adaptive estimation,
Empirical processes, Sieves, Minimum contrast estimators.

**Classification MSC :** primary 62G05, 62G07, secondary 41A25

**Résumé :**

- Performance bounds for criteria for Model Selection are developed using recent theory for sieves. The model selection criteria are based on an empirical loss or contrast function with an added penalty term motivated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of observations. Most of our examples involve regression or density estimation settings and we focus on the problem of estimating the unknown density or regression function. We show that the quadratic risk of the "penalized minimum contrast estimator" is bounded by an index of the accuracy of the sieve. This accuracy index quantifies the trade-off among the candidate models between the approximation error and parameter dimension relative to sample size. If we choose a list of models which exhibit good approximation properties with respect to different classes of smoothness, the estimator can be simultaneously minimax rate optimal in each of those classes. This is what is usually called {\em adaptation}. The type of classes of smoothness in which one gets adaptation depends heavily on the list of models. If too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously, it may happen that the estimator is only approximately adaptive (typically up to a slowly varying function of the sample size). We shall provide various illustrations of our methods such as penalized maximum likelihhood, projection or least squares estimation. The models will involve commonly used finite dimensional expansions such as piecewise polynomials with fixed or variable knots, trigonometric polynomials, wavelets, neural nets and related nonlinear expansions defined by superposition of ridge functions.

**Article :
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**Contact :
**Pascal.Massart@math.u-psud.fr