Bounds for Bayesian Order Identification with Application to Mixtures

The efficiency of two Bayesian order estimators is studied. By using nonparametric techniques, we prove new underestimation and overestimation bounds. The results apply to various models, including mixture models. In this case, the errors are shown to be O(e −an) and O((log n) b / √ n) (a, b > 0), respectively. 1. Introduction. Order identification deals with the estimation and test of a structural parameter which indexes the complexity of a model. In other words, the most economical representation of a random phenomenon is sought. This problem is encountered in many situations, including: mixture models [13, 19] with an unknown number of components; cluster analysis [9], when the number of clusters is unknown; autoregressive models [1], when the process memory is not known. This paper is devoted to the study of two Bayesian estimators of the order of a model. Frequentist properties of efficiency are particularly investigated. We obtain new efficiency bounds under mild assumptions, providing a theoretical answer to the questions raised, for instance, in [7] (see their Section 4).