Spatial modeling using graphical Markov models and wavelets

Abstract

Partially ordered Markov models (POMMs) are Markov random fields (MRPs) with neigh­ borhood structures derivable from an associated partially ordered set. The most attractive feature of POMMs is that their joint distributions can be written in closed and product form. Therefore, simulation and maximum likelihood estimation for the models is quite straightfor­ ward, which is not the case in general for MRP models. In practice, one often has to modify the likelihood to account for edge components; the resulting composite likelihood for POMMs is similarly straightforward to maximize. In this article, we use a martingale approach to derive the asymptotic properties of maximum (composite) likelihood estimators for POMMs. One of our results establishes that, under regularity conditions that include Dobrushin's condition for spatial mixing, the maximum composite likelihood estimator is consistent, asymptotically normal, and also asymptotically efficient. AMS 1991 subject classifications: Primary 62M40; secondary 62F12.