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Goodness of fit tests for a class of Markov random field models. (English) Zbl 1246.62179
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.

62M02 Markov processes: hypothesis testing
62M30 Inference from spatial processes
62M40 Random fields; image analysis
65C60 Computational problems in statistics (MSC2010)
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[1] Anderson, T. W. (1993). Goodness of fit tests for spectral distributions. Ann. Statist. 21 830-847. · Zbl 0779.62083
[2] Arnold, B. C., Castillo, E. and Sarabia, J. M. (1992). Conditionally Specified Distributions. Lecture Notes in Statistics 73 . Springer, Berlin. · Zbl 0749.62002
[3] Bai, J. (2003). Testing parametric conditional distributions of dynamic models. Rev. Econom. Statist. 85 531-549.
[4] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192-236. · Zbl 0327.60067
[5] Besag, J. and Higdon, D. (1999). Bayesian analysis of agricultural field experiments. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 691-746. · Zbl 0951.62091
[6] Besag, J. and Kooperberg, C. (1995). On conditional and intrinsic autoregressions. Biometrika 82 733-746. · Zbl 0899.62123
[7] Brockwell, A. E. (2007). Universal residuals: A multivariate transformation. Statist. Probab. Lett. 77 1473-1478. · Zbl 1128.62064
[8] Caragea, P. C. and Kaiser, M. S. (2009). Autologistic models with interpretable parameters. J. Agric. Biol. Environ. Stat. 14 281-300. · Zbl 1306.62255
[9] Cox, D. R. and Snell, E. J. (1971). On test statistics calculated from residuals. Biometrika 58 589-594. · Zbl 0228.62017
[10] Cressie, N. A. C. (1993). Statistics for Spatial Data , 2nd ed. Wiley, New York. · Zbl 0799.62002
[11] Csiszár, I. and Talata, Z. (2006). Consistent estimation of the basic neighborhood of Markov random fields. Ann. Statist. 34 123-145. · Zbl 1102.62105
[12] Czado, C., Gneiting, T. and Held, L. (2009). Predictive model assessment for count data. Biometrics 65 1254-1261. · Zbl 1180.62162
[13] Darling, D. A. (1957). The Kolmogorov-Smirnov, Cramér-von Mises tests. Ann. Math. Statist. 28 823-838. · Zbl 0082.13602
[14] Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Series in Statistical and Probabilistic Mathematics 1 . Cambridge Univ. Press, Cambridge. · Zbl 0886.62001
[15] Dawid, A. P. (1984). Statistical theory. The prequential approach. J. Roy. Statist. Soc. Ser. A 147 278-292. · Zbl 0557.62080
[16] Diebold, F. X., Gunther, T. A. and Tay, A. S. (1998). Evaluating density forecasts with applications to financial risk management. Internat. Econom. Rev. 39 863-883.
[17] Durbin, J. (1973). Weak convergence of the sample distribution function when parameters are estimated. Ann. Statist. 1 279-290. · Zbl 0256.62021
[18] Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457-511. · Zbl 1386.65060
[19] Gneiting, T., Balabdaoui, F. and Raftery, A. E. (2007). Probabilistic forecasts, calibration and sharpness. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 243-268. · Zbl 1120.62074
[20] Guyon, X. (1995). Random Fields on a Network . Springer, New York. · Zbl 0839.60003
[21] Guyon, X. and Yao, J.-F. (1999). On the underfitting and overfitting sets of models chosen by order selection criteria. J. Multivariate Anal. 70 221-249. · Zbl 1070.62516
[22] Hammersley, J. M. and Clifford, P. (1971). Markov fields on finite graphs and lattices. Unpublished manuscript.
[23] Hardouin, C. and Yao, J.-F. (2008). Multi-parameter auto-models with applications to cooperative systems and analysis of mixed state data. Biometrika 95 335-349. · Zbl 1437.62490
[24] Jager, L. and Wellner, J. A. (2007). Goodness-of-fit tests via phi-divergences. Ann. Statist. 35 2018-2053. · Zbl 1126.62030
[25] Jensen, T. R. and Toft, B. (1995). Graph Coloring Problems . Wiley, New York. · Zbl 0855.05054
[26] Ji, C. and Seymour, L. (1996). A consistent model selection procedure for Markov random fields based on penalized pseudolikelihood. Ann. Appl. Probab. 6 423-443. · Zbl 0856.62082
[27] Justel, A., Peña, D. and Zamar, R. (1997). A multivariate Kolmogorov-Smirnov test of goodness of fit. Statist. Probab. Lett. 35 251-259. · Zbl 0883.62054
[28] Kaiser, M. S. and Caragea, P. C. (2009). Exploring dependence with data on spatial lattices. Biometrics 65 857-865. · Zbl 1172.62031
[29] Kaiser, M. S. and Cressie, N. (1997). Modeling Poisson variables with positive spatial dependence. Statist. Probab. Lett. 35 423-432. · Zbl 0904.62067
[30] Kaiser, M. S. and Cressie, N. (2000). The construction of multivariate distributions from Markov random fields. J. Multivariate Anal. 73 199-220. · Zbl 1065.62520
[31] Kaiser, M. S., Cressie, N. and Lee, J. (2002). Spatial mixture models based on exponential family conditional distributions. Statist. Sinica 12 449-474. · Zbl 0998.62079
[32] Kaiser, M. S., Lahiri, S. N. and Nordman, D. J. (2011). Supplement to “Goodness of fit tests for a class of Markov random field models.” . · Zbl 1246.62179
[33] Khmaladze, È. V. (1981). A martingale approach in the theory of goodness-of-fit tests. Theory Probab. Appl. 26 240-257. · Zbl 0481.60055
[34] Khmaladze, È. V. (1993). Goodness of fit problem and scanning innovation martingales. Ann. Statist. 21 798-829. · Zbl 0801.62043
[35] Khmaladze, E. V. and Koul, H. L. (2004). Martingale transforms goodness-of-fit tests in regression models. Ann. Statist. 32 995-1034. · Zbl 1092.62052
[36] Koul, H. L. (1970). A class of ADF tests for subhypothesis in the multiple linear regression. Ann. Math. Statist. 41 1273-1281. · Zbl 0202.17101
[37] Koul, H. L. and Sakhanenko, L. (2005). Goodness-of-fit testing in regression: A finite sample comparison of bootstrap methodology and Khmaladze transformation. Statist. Probab. Lett. 74 290-302. · Zbl 1070.62030
[38] Lahiri, S. N. (1999). Asymptotic distribution of the empirical spatial cumulative distribution function predictor and prediction bands based on a subsampling method. Probab. Theory Related Fields 114 55-84. · Zbl 0951.62013
[39] Lahiri, S. N. (2003). Central limit theorems for weighted sums of a spatial process under a class of stochastic and fixed designs. Sankhyā Ser. A 65 356-388. · Zbl 1192.60054
[40] Lahiri, S. N., Kaiser, M. S., Cressie, N. and Hsu, N.-J. (1999). Prediction of spatial cumulative distribution functions using subsampling. J. Amer. Statist. Assoc. 94 86-110. With comments and a rejoinder by the authors. · Zbl 1180.62133
[41] Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23 470-472. · Zbl 0047.13104
[42] Rue, H. and Held, L. (2005). Gaussian Markov Random Fields : Theory and Applications. Monographs on Statistics and Applied Probability 104 . Chapman and Hall/CRC, Boca Raton, FL. · Zbl 1093.60003
[43] Sherman, M. and Carlstein, E. (1994). Nonparametric estimation of the moments of a general statistic computed from spatial data. J. Amer. Statist. Assoc. 89 496-500. · Zbl 0798.62047
[44] Smith, R. L. (1999). Discussion of “Bayesian analysis of agricultural field experiments,” by J. Besag and D. Higdon. J. Roy. Statist. Soc. Ser. B 61 725-727. · Zbl 0951.62091
[45] Speed, T. P. (1978). Relations between models for spatial data, contingency tables and Markov fields on graphs. Suppl. Adv. Appl. Probab. 10 111-122. · Zbl 0388.62056
[46] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
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