GOODNESS OF FIT TESTS FOR A CLASS OF MARKOV RANDOM FIELD MODELS

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 g...

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Bibliographic Details
Published in:The Annals of statistics 2012-02, Vol.40 (1), p.104-130
Main Authors: Kaiser, Mark S., Lahiri, Soumendra N., Nordman, Daniel J.
Format: Article
Language:English
Subjects:
62F03
62M30
Goodness of fit
Increasing domain asymptotics
Markov analysis
Markov models
Neighborhoods
Null hypothesis
Perceptual localization
probability integral transform
Random variables
Sampling distributions
Simulation
Spatial data
Spatial models
spatial processes
spatial residuals
Statistical theories
Statistics
Studies
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/11-AOS948