Asymptotic Comparison of Estimators in the Ising Model

Because of their use as priors in image analysis, the interest in parameter estimation for Gibbs random fields has rosen recently. Gibbs fields form an exponential family, so maximum likelihood would be the estimator of first choice. Unfortunately it is extremly difficult to compute. Other estimators which are easier to compute have been proposed: the Coding and the pseudo-maximum likelihood estimator (Besag, 1974), a minimum chi-square estimator (Glotzl and Rauchenschwandtner, 1981; Possolo, 1986-a) and the conditional least squares estimator (Lele et Ord, 1986), cf the definitions below in section 2.2. - These estimators are all known to be consistent. Hence it is a natural question to compare efficiency among these simple estimators and with respect to the maximum likehood estimator. We do this here in the simplest non trivial case, the d-dimensional nearest neighbor isotropic Ising model with external field. We show that both the pseudo maximum likelihood and the conditional least squares estimator are asymptotically equivalent to a minimum chi-square estimator when the weight matrix for the latter is chosen appropriately (corollary 2). These weight matrices are different from the optimal matrix. Hence we expect also the resulting estimators to be different although in all our examples the maximum pseudo likelihood and the minimum chi-square estimator with optimal weight turned out to be asymptotically equivalent. In particular, our results do not confirm the superior behavior of minimum chi-square over pseudo maximun likehood reported in Possolo (1986a). By example, we show that conditional least squares and minimum chi-square with the identity matrix as weights can be worse than the optimal minimum chi-square estimator. Compared with the maximum likelihood, the easily computable estimators are not bad if the interaction is weak, but much worse if the interaction is strong. Our results suggest that their asymptotic efficiency tends to zero as one approaches the critical point.