Asymptotic Theory of Cepstral Random Fields

McElroy, T.S., and S.H. Holan. Asymptotic Theory of Cepstral Random Fields. University of Missouri Preprint 1813:34461, 2012, available at
Asymptotic Theory of Cepstral Random Fields McElroy, T.S.; Holan, S.H. 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. Given the importance of this topic, there has been a substantial amount of research devoted to this area. However, the cepstral random field model remains largely underdeveloped outside the engineering literature. We provide a comprehensive treatment of the asymptotic theory for two-dimensional random field models. In particular, we provide recursive formulas that connect the spatial cepstral coefficients to an equivalent moving-average random field, which facilitates easy computation of the necessary autocovariance matrix. Additionally, we establish asymptotic consistency results for Bayesian, maximum likelihood, and quasi-maximum likelihood estimation of random field parameters and regression parameters. Further, in both the maximum and quasi-maximum likelihood frameworks, we derive the asymptotic distribution of our estimator. 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. Finally, our results are illustrated through simulation and the analysis of straw yield data in an agricultural field experiment.