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Consistency of objective Bayes factors as the model dimension grows. (English) Zbl 1323.62024

Summary: In the class of normal regression models with a finite number of regressors, and for a wide class of prior distributions, a Bayesian model selection procedure based on the Bayes factor is consistent [G. Casella and E. Moreno, J. Am. Stat. Assoc. 104, No. 487, 1261–1271 (2009; Zbl 1328.62359)]. However, in models where the number of parameters increases as the sample size increases, properties of the Bayes factor are not totally understood. Here we study consistency of the Bayes factors for nested normal linear models when the number of regressors increases with the sample size. We pay attention to two successful tools for model selection [G. Schwarz, Ann. Stat. 6, 461–464 (1978; Zbl 0379.62005)] approximation to the Bayes factor, and the Bayes factor for intrinsic priors [J. O. Berger and L. R. Pericchi, J. Am. Stat. Assoc. 91, No. 433, 109–122 (1996; Zbl 0870.62021); E. Moreno et al., ibid. 93, No. 444, 1451–1460 (1998; Zbl 1064.62513)].
We find that the the Schwarz approximation and the Bayes factor for intrinsic priors are consistent when the rate of growth of the dimension of the bigger model is \(O(n^b)\) for \(b < 1\). When \(b = 1\) the Schwarz approximation is always inconsistent under the alternative while the Bayes factor for intrinsic priors is consistent except for a small set of alternative models which is characterized.

MSC:

62F05 Asymptotic properties of parametric tests
62J15 Paired and multiple comparisons; multiple testing
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References:

[1] Berger, J. O., Ghosh, J. K. and Mukhopadhyay, N. (2003). Approximations and consistency of Bayes factors as model dimension grows. J. Statist. Plann. Inference 112 241-258. · Zbl 1026.62018
[2] Berger, J. O. and Pericchi, L. R. (1996). The intrinsic Bayes factor for model selection and prediction. J. Amer. Statist. Assoc. 91 109-122. JSTOR: · Zbl 0870.62021
[3] Casella, G., Girón, F. J., Martínez, M. L. and Moreno, E. (2009). Consistency of Bayesian procedures for variable selection. Ann. Statist. 37 1207-1228. · Zbl 1160.62004
[4] Casella, G. and Moreno E. (2006). Objective Bayesian variable selection. J. Amer. Statist. Assoc. 101 157-167. · Zbl 1118.62313
[5] Casella, G. and Moreno E. (2009). Assessing robustness of intrinsic test of independence in two-way contingency tables. J. Amer. Statist. Assoc. 104 1261-1271. · Zbl 1328.62359
[6] Girón, F. J., Martínez, M. L., Moreno, E. and Torres, F. (2006). Objective testing procedures in linear models. Calibration of the p -values. Scand. J. Statist. 33 765-784. · Zbl 1164.62322
[7] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions , 2nd ed. Wiley, New York. · Zbl 0821.62001
[8] Liang F., Paulo, R., Molina G., Clyde M. and Berger J. O. (2008). Mixtures of g -priors for Bayesian Variable Selection. J. Amer. Statist. Assoc. 103 410-423. · Zbl 1335.62026
[9] Moreno, E., Bertolino, F. and Racugno, W. (1998). An intrinsic limiting procedure for model selection and hypothesis testing. J. Amer. Statist. Assoc. 93 1451-1460. JSTOR: · Zbl 1064.62513
[10] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461-464. · Zbl 0379.62005
[11] Shao, J. (1997). An asymptotic theory for linear model selection. Statist. Sinica 7 221-264. · Zbl 1003.62527
[12] Stone, M. (1979). Comments on model selection criteria of Akaike and Schwarz. J. Roy. Statist. Soc. Ser. B 41 276-278.
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