  Random fields play a central role in the analysis of spatially correlated
data and, as a result, have a significant impact on a broad array of scientific
applications. This paper studies the cepstral random field model, providing
recursive formulas that connect the spatial cepstral coefficients to an
equivalent moving-average random field, which facilitates easy computation of
the autocovariance matrix. We also provide a comprehensive treatment of the
asymptotic theory for two-dimensional random field models: we establish
asymptotic results for Bayesian, maximum likelihood and quasi-maximum
likelihood estimation of random field parameters and regression parameters. The
theoretical results are presented generally and are of independent interest,
pertaining to a wide class of random field models. The results for the cepstral
model facilitate model-building: because the cepstral coefficients are
unconstrained in practice, numerical optimization is greatly simplified, and we
are always guaranteed a positive definite covariance matrix. We show that
inference for individual coefficients is possible, and one can refine models in
a disciplined manner. Our results are illustrated through simulation and the
analysis of straw yield data in an agricultural field experiment.
