Consistent Order Estimation and the Local Geometry of Mixtures by Elisabeth Gassiat

Consider an i.i.d. sequence of random variables whose distribution f lies in one of a nested family of models (Mq)q∈N, Mq ⊂ Mq+1. The smallest index q such that Mq⋆ contains f is called the model order. We establish strong consistency of the penalized likelihood order estimator in a general setting with penalties of order η(q) log log n, where η(q) is a dimensional quantity. Moreover, such penalties are shown to be minimal. In contrast to previous work, an a priori upper bound on the model order is not assumed. The local dimension η(q) of the model Mq is defined in terms of the bracketing entropy of a class of weighted densities, whose computation is a nonstandard problem which is of independent interest. We perform the requisite computations for the case of one-dimensional location mixtures, thus demonstrating the consistency of the penalized likelihood mixture order estimator. The proof requires a delicate analysis of the local geometry of the mixture family Mq in a neighborhood of f, for q > q. The extension to more general mixture models remains an open problem.