Consistency of modified LS estimation method for identifying 2-D noncausal SAR model parameters

There are two main methods for estimating the parameters of two-dimensional (2-D) noncausal simultaneous autoregressive (SAR) models. One is the least-squares (LS) and the other is the maximum likelihood (ML). The asymptotically unbiased and consistent ML, method is computationally unattractive even after some approximations have been introduced. On the other hand, the computationally efficient conventional LS method does not produce accurate parameter estimates in this case due to the noncausality of the models. In order to improve the estimation accuracy and keep the computational efficiency of the LS method, an unbiased modified LS estimator was recently proposed. However, a very important matter remains to be addressed. As of yet, a mathematical proof for the consistency of the modified LS estimator has not been presented anywhere in the literature. The results of previous computer simulation studies on this estimator have been based on only data sample windows of fixed sizes. The studies are limited by the fact that the variances and mean square errors of the parameter estimates as functions of the data window sizes could not be deduced from the results. Therefore, the results presented to date cannot be used to demonstrate either the consistency or the convergence properties of the estimator. Based on both analytical and experimental investigations, this paper proves the consistency of the modified LS estimator. A detailed theoretical analysis and a new numerical example are included in the paper. The experimental results corroborate the theoretical results