CHAMBAZ, Antoine - Modélisation Stochastique et Statistique, Université Paris-Sud, Bât. 425, 91405 Orsay cedex

Abstract :

We study in this paper some asymptotic properties of joint procedures of estimation and test of the order of a model. Given a collection of nested parametric models and a distribution in their union, the order of the latter is the order of the smallest model the distribution belongs to. It is a number that quantifies the complexity of the model regarding the whole collection. We define two estimators which yield two test procedures. We prove under mild assumptions their almost sure consistency, possibly without any prior bound on the order of interest $\Kstar$. We also cope with their rates of under- and overestimation ({\ie} when the estimated order is lower or larger than the true one, respectively). These rates respectively bound above the errors of second and first kind when testing $\Kstar\leq K_{0}$ {\vs} $\Kstar> K_{0}$. We get under mild assumptions a universal lower bound for the underestimation rate as well as upper bounds which all exponentially decrease with the sample size $n$. We prove that the best possible rate in exponential models is achieved. Finally, we show under mild assumptions that the overestimation rate is necessarily slower than exponential in $n$ and get exponential rates in $n^{1-\delta}$ (any $\delta\in]0,1[$). The originality of our approach relies in the substitution of functional arguments to explicit calculus while dealing with maximum likelihood. This confers greater generality on our results. Some benchmark examples (mixture of Gaussians, abrupt changes and various regressions) are carefully addressed.

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Contact : Antoine.Chambaz@math.u-psud.fr