102
28

Consistent order estimation and the local geometry of mixtures

Abstract

Consider an i.i.d. sequence of random variables whose distribution ff^\star lies in one of a nested family of models (Mq)qN(\mathcal{M}_q)_{q\in\mathbb{N}}, MqMq+1\mathcal{M}_q\subset \mathcal{M}_{q+1}. The smallest index qq^\star such that Mq\mathcal{M}_{q^\star} contains ff^\star 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)loglogn\eta(q)\log\log n, where η(q)\eta(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)\eta(q) of the model Mq\mathcal{M}_q 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\mathcal{M}_q in a neighborhood of ff^\star, for q>qq>q^\star. The extension to more general mixture models remains an open problem.

View on arXiv
Comments on this paper

We use cookies and other tracking technologies to improve your browsing experience on our website, to show you personalized content and targeted ads, to analyze our website traffic, and to understand where our visitors are coming from. See our policy.