Feasible Asymptotic Inference in Nonlinear Random Fields: CLT, Empirical Information Operators, and Spatial Application

In this paper, we establish a central limit theorem (CLT) for the maximum likelihood estimator (MLEs) in two-dimensional random coefficient autoregressive (2D-RCAR) models. We show that the normalized estimation error converges in distribution to a centered normal law, with an asymptotic covariance matrix explicitly determined. This theoretical result provides a rigorous foundation for asymptotic inference. We propose empirical estimators of this covariance structure, enabling the construction of confidence intervals and hypothesis tests for the model parameters. The validity of the normal approximation is confirmed through simulation results.