Statistics and Its Interface

Volume 5 (2012)

Number 3

Blockwise empirical likelihood for spatial Markov model assessment

Pages: 303 – 318



Mark S. Kaiser (Department of Statistics, Iowa State University, Ames, Ia., U.S.A.)

Daniel J. Nordman (Department of Statistics, Iowa State University, Ames, Ia., U.S.A.)


We propose a class of procedures for the assessment of Markov random field models based on spatial blockwise empirical likelihood (SBEL). These SBEL procedures have discernible asymptotic properties and provide a means for testing particular assumptions made in model formulation, such as distributional form and neighborhood structure. Based on the ideas of empirical likelihood, SBEL procedures are nonparametric in that they do not depend on parameterized likelihood functions per se. However, the moment conditions that are the focus of SBEL tests may themselves reflect data behaviors that are dictated by specific components of fully parametric models. Model assessment based on SBEL, then, produces completely data-driven procedures for testing the assumptions of parameterized Markov random field models. The procedures require two sets of moment conditions that are formulated as estimating functions. One set of estimating functions provides identification of parameter values and the other serves to define aspects of model behavior that we wish to assess relative to the behavior of observed data. We illustrate the use of SBEL procedures using null (assumed) models that have Gaussian and binary conditional distributions when data are simulated from both these, and other, models. Among other results, we demonstrate that an appropriate SBEL procedure is capable of detecting incongruence between assumed and true neighborhood structure on a regular lattice.


estimating functions, Markov random fields, pseudo-likelihood

2010 Mathematics Subject Classification

Primary 62G09, 62M30. Secondary 62F03.

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