Specification of dependence structures and simulation-based estimation for conditionally specified statistical models

摘要

Conditionally specified statistical models are frequently constructed from conditional one-parameter exponential family distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field. When this is done, a common assumption is that dependence is expressed only through pairs of random variables, the \u27pairwise-only dependence\u27 assumption. Using a Markov random field structure and the pairwise-only dependence assumption, Besag (1974) formulated exponential family \u27auto-models\u27, and showed the form that conditional one-parameter exponential family densities must have in such models. Those results are extended under relaxation of the pairwise-only dependence assumption, and a necessary form for conditional one-parameter exponential family densities is given under more general conditions of multiway dependence;A strategy is proposed for maximum likelihood estimation of parameters appearing in the joint distribution of a set of random variables modeled through the specification of full conditional probability density or mass functions. This strategy relies on maximization of a sequence of Monte Carlo approximations to the log likelihood function. The fundamental issue addressed in our strategy is formulation of an importance sampling distribution as a product of marginal functions, where those marginals are chosen in a way that reflects the influence of dependence on the first two moments of the actual statistical model under consideration. We address a number of practical issues in the use of Monte Carlo methods to locate maximum likelihood estimates, including criteria for when an additional sampling distribution should be selected and the selection of appropriate starting values. This estimation strategy is extended to mixture models in which the mixing distributions are identified up to the normalizing constants by the specification of full conditional probability density or mass functions;The large sample theory for the resulting estimates from the proposed strategy is provided under the condition of the continuity of the negpotential function over the compact set. In addition, convergence of the Monte Carlo estimate of log likelihood to the true log likelihood and asymptotic results are given for the theoretical support for one of solutions of practical issues in the estimation strategy.