Model selection by the Schwarz criterion and the resampling technique. Application to Gaussian Mixture Models

In the selection of the order of a model by information criteria, the estimator of the order obtained by the Schwarz Information Criterion $(BIC)$ \cite{Ref9} is convergent almost surely. In practice, the selection by this criterion generally leads to a underparametrization of the model for small samples. To overcome this problem, Chen et al. \cite{Ref2, Ref3} have proposed a modification of the criterion $BIC$ by the resampling technique for autoregressive models. This new criterion generally allows to decrease the underparametrization and they have shown that it preserves the convergence property of the order estimator. In this paper, we propose a new criterion, denoted $BIC_r$, generalizing the results of Chen et al. to parametric models. We apply this technique to estimate the order of a Gaussian Mixture Model (GMM), and we illustrate the performance of our method on simulated data set.