Time-invariance Estimating Equations

We describe a general method for deriving estimators of the parameter of a statistical model, with particular relevance to highly structured stochastic systems such as conditional independence models and random spatial processes. The method is based on representing the stochastic model as the equilibrium distribution of a continuous-time Markov process Y = (Y t ; t > 0) wherètime' is to be understood as a ctional extra dimension added to the original setting. The parameter estimate ^ T is obtained by equating to zero the innnitesimal generator of Y applied to a suitable statistic and evaluated at the data x. This produces an unbiased estimating equation for. Special cases include Besag's maximum pseudolikelihood estimator for discrete Gibbs random elds and for Gibbs point processes, the Takacs-Fiksel method for point processes, some of thèvariational estimators' of Almeida and Gidas, and some moment estimators in stochastic geometry. In a general setting this approach also embraces the method of moments, and maximum likelihood for exponential families. It has some aanity with the Stein-Chen method for distributional approximation.