Time-invariance estimating equations

Abstract

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 spatial random processes and `graphical' conditional independence models. The method is based on representing the stochastic model $X$ as the equilibrium distribution of an auxiliary Markov process $Y=(Y_t,t>0)$ where the discrete or continuous 'time' index t is to be understood as a fictional extra dimension added to the original setting. The parameter estimate $\hat{\theta }$ is obtained by equating to zero the generator of $Y$ applied to a suitable statistic and evaluated at the data $x$ . This produces an unbiased estimating equation for θ. Natural special cases include maximum likelihood, the method of moments, the reduced sample estimator in survival analysis, the maximum pseudolikelihood estimator for random fields and for point processes, the Takacs-Fiksel method for point processes, 'variational' estimators for random fields and multivariate distributions, and many standard estimators in stochastic geometry. The approach has some affinity with the Stein-Chen method for distributional approximation.