The Annals of Statistics

Asymptotic theory of cepstral random fields

Tucker S. McElroy and Scott H. Holan

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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.

Article information

Ann. Statist., Volume 42, Number 1 (2014), 64-86.

First available in Project Euclid: 15 January 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62M30: Spatial processes
Secondary: 62F15: Bayesian inference 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Bayesian estimation cepstrum exponential spectral representation lattice data spatial statistics spectral density


McElroy, Tucker S.; Holan, Scott H. Asymptotic theory of cepstral random fields. Ann. Statist. 42 (2014), no. 1, 64--86. doi:10.1214/13-AOS1180.

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Supplemental materials

  • Supplementary material: Supplement to asymptotic theory of cepstral random fields. The supplement contains a description of further applications of the cepstral model, analysis of straw yield data, as well as all proofs.