The Annals of Statistics

Goodness of fit tests for a class of Markov random field models

Mark S. Kaiser, Soumendra N. Lahiri, and Daniel J. Nordman

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This paper develops goodness of fit statistics that can be used to formally assess Markov random field models for spatial data, when the model distributions are discrete or continuous and potentially parametric. Test statistics are formed from generalized spatial residuals which are collected over groups of nonneighboring spatial observations, called concliques. Under a hypothesized Markov model structure, spatial residuals within each conclique are shown to be independent and identically distributed as uniform variables. The information from a series of concliques can be then pooled into goodness of fit statistics. Under some conditions, large sample distributions of these statistics are explicitly derived for testing both simple and composite hypotheses, where the latter involves additional parametric estimation steps. The distributional results are verified through simulation, and a data example illustrates the method for model assessment.

Article information

Ann. Statist. Volume 40, Number 1 (2012), 104-130.

First available in Project Euclid: 15 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F03: Hypothesis testing
Secondary: 62M30: Spatial processes

Increasing domain asymptotics probability integral transform spatial processes spatial residuals


Kaiser, Mark S.; Lahiri, Soumendra N.; Nordman, Daniel J. Goodness of fit tests for a class of Markov random field models. Ann. Statist. 40 (2012), no. 1, 104--130. doi:10.1214/11-AOS948.

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Supplemental materials

  • Supplementary material: Proofs of main results for spatial GOF test statistics. A supplement [32] provides proofs of all asymptotic distributional results from Section 4, regarding the conclique-based spatial GOF test statistics in simple and composite null hypothesis settings (Proposition 4.1, Theorem 4.2, Corollary 4.3, Theorem 4.4, Corollary 4.5). The proof in the composite hypothesis case is particularly nonstandard; see Section 4.4.