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Objective Bayesian analysis for CAR models. (English) Zbl 1396.62044

Summary: Objective priors, especially reference priors, have been studied extensively for spatial data in the last decade. In this paper, we study objective priors for a CAR model. In particular, the properties of the reference prior and the corresponding posterior are studied. Furthermore, we show that the frequentist coverage probabilities of posterior credible intervals depend only on the spatial dependence parameter \(\rho \), and not on the regression coefficient or the error variance. Based on the simulation study for comparing the reference and Jeffreys priors, the performance of two reference priors is similar and better than the Jeffreys priors. One spatial dataset is used for illustration.

MSC:

62F15 Bayesian inference
62M30 Inference from spatial processes

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References:

[1] Bell, B. S., Broemeling, L. D. (2000). A Bayesian analysis of spatial processes with application to disease mapping. Statistics in Medicine, 19, 957-974.
[2] Berger, J. O., Bernardo, J. M. (1992). On the development of reference priors (Disc: p49-60). In Bayesian statistics 4: Proceedings of the fourth Valencia international meeting (pp. 35-49).
[3] Berger, J. O., Philippe, A., Robert, C. (1998). Estimation of quadratic functions: Noninformative priors for non-centrality parameters. Statistica Sinica, 8, 359-376. · Zbl 0899.62037
[4] Berger, J. O., De Oliveira, V., Sansó, B. (2001). Objective Bayesian analysis of spatially correlated data. Journal of the American Statistical Association, 96, 1361-1374. · Zbl 1051.62095
[5] Besag, J. G. P. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B, Methodological, 36, 3-66. · Zbl 0327.60067
[6] Clayton, D., Kaldor, J. (1987). Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics, 43, 671-681.
[7] Cressie, N. A. C. (1993). Statistics for spatial data (rev ed.). New York: Wiley. · Zbl 0799.62002
[8] Cressie, N. A. C., Chan, N. H. (1989). Spatial modeling of regional variables. Journal of the American Statistical Association, 84, 393-401. · Zbl 1248.62211
[9] Cressie, N., Perrin, O., Thomas-Agnan, C. (2005). Likelihood-based estimation for Gaussian mrfs. Statistical Methodology, 2, 1-16. · Zbl 1248.62171
[10] Datta, G. S., Ghosh, M. (1996). On the invariance of noninformative priors. The Annals of Statistics, 24, 141-159. · Zbl 0906.62024
[11] De Oliveira, V. (2012). Bayesian analysis of conditional autoregressive models. Annals of the Institute of Statistical Mathematics, 64, 107-133. · Zbl 1238.62028
[12] Harville, D. A. (1974). Bayeisan inference for variance components using only error contrasts. Biometrika, 61, 383-385. · Zbl 0281.62072
[13] Paulo, R. (2005). Default priors for Gaussian processes. The Annals of Statistics, 33, 556-582. · Zbl 1069.62030
[14] Ren, C., Sun, D., He, Z. (2012). Objective Bayesian analysis for a spatial model with nugget effects. Journal of Statistical Planning and Inference, 142, 1933-1946. · Zbl 1237.62034
[15] Richardson, S., Guihenneuc, C., Lasserre, V. (1992). Spatial linear models with autocorrelated error structure. The Statistician, 41, 539-557.
[16] Rue, H., Held, L. (2005). Gaussian Markov random fields: Theory and applications. Boca Raton: Chapman and Hall/CRC. · Zbl 1093.60003
[17] Sun, D., Ye, K. (1995). Reference prior Bayesian analysis for normal mean products. Journal of the American Statistical Association, 90, 589-597. · Zbl 0833.62030
[18] Wakefield, J. C., Gelfand, A. E., Smith, A. F. M. (1991). Efficient generation of random variates via the ratio-of-uniforms method. Statistics and Computing, 1, 129-133.
[19] Welch, B., Peers, H. W. (1963). Reference prior Bayesian analysis for normal mean products. Journal of the Royal Statistical Society B, 25, 318-329. · Zbl 0117.14205
[20] Ye, K. (1994). Bayesian reference prior analysis on the ratio of variances for the balanced one-way random effect model. Journal of Statistical Planning and Inference, 41, 267-280. · Zbl 0816.62029
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