Abstract: Partially ordered Markov models (POMMs) are Markov random fields (MRFs) with neighborhood 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 straightforward, which is not the case in general for MRF 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 are fairly easy to check, and Dobrushin's condition for spatial mixing, the maximum composite likelihood estimator is consistent, asymptotically normal, and also asymptotically efficient.
Key words and phrases: Acyclic directed graph, asymptotic efficiency, asymptotic normality, consistency, Dobrushin's condition, level set, Markov random field, martingale central limit theorem, strong mixing, triangular martingale array.