Nonconjugate Bayesian Estimation of Covariance Matrices and its Use in Hierarchical Models

Journal of the American Statistical Association

Volumn 94, Issue 448, 1999, Pages 1254-1263

  Daniels, Michael J ^a   Kass, Robert E ^b  

^a Department of Statistics Center for Statistics and Applications in Forensic Evidence Iowa State University 2438 Osborn Dr Ames   (United States)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Givens angles; Hierarchical prior; Random effects; Shrinkage; Variance matrix

Indexed keywords

EID: 0442325081     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.1999.10473878     Document Type: Article

References (30)

1. Barnard, J., McCulloch, R., and Mcng, X. (1996), "A Natural Strategy for Modeling Covariance Matrices With Application to Shrinkage," technical report. Harvard University, Dept, of Statistics.
2. Bennet, J. E, Racine-Poon, A., and Wakefield, J. C. (1995), "MCMC for Nonlinear Hierarchical Models," in Markov Chain Monte Carlo in Practice, eds. W. R. Gilks, S. Richardson, and D. J. Spiegclhalter, New York: Chapman and Hall, pp. 339-358.
3. Berger, J., and Bernardo, J. M. (1992), "On the Development of Reference Priors" (with discussion), in Bayesian Statistics 4, eds. J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, Oxford, U.K.: Oxford University Press, pp. 35-60.
4. Christiansen, C. L., and Morris, C. N. (1997), "Hierarchical Poisson Re-gression Modeling," Journal of the American Statistical Association, 92, 618-632.
5. Daniels, M. J. (1998), "Computing Posterior Distributions for Covariance Matrices," Computing Science and Statistics, 30, ed. S. Weisberg, pp. 192-196.
6. Daniels, M. J. (1999), "A Prior for the Variance in Hierarchical Models," Canadian Journal of Statistics, to appear.
7. Daniels, M. J. and Crcssie, N. (1998), "A Hierarchical Approach to Covariance Function Estimation in Time Series," submitted.
8. Daniels, M. J. and Crcssie, N and Gatsonis, C. (1999), "Hierarchical Generalized Linear Models in the Analysis of Variations in Health Care Utilization," Journal of the American Statistical Association, 94, 29-42.
9. Dey, D. K and Srinivasan, C. (1985), "Estimation of a Covariance Matrix Under Stein's Loss," The Annals of Statistics, 13, 1581-1591.
10. Everson, P. J., and Morris, C. N. (1997), "Inference for Multivariate Normal Hierarchical Models," Journal of the Royal Statistical Society, Series B, to appear.
11. Genz, A., and Kass, R. E. (1997), "Subregion-Adaptive Integration of Functions Having a Dominant Peak," Journal of Computational and Graphical Statistics, 6, 92-111.
12. Gilks, W. R., Richardson, S., and Spiegelhalter, D. J. (1995), Markov Chain Monte Carlo in Practice, London: Chapman and Hall.
13. Goldstein, H. (1962), Classical Mechanics, Reading, MA: Addison-Wesley.
14. Haff, L. R. (1991), 'The Variational Form of Certain Bayes Estimators," The Annals of Statistics, 19, 1163-1190.
15. Hobert, J. P., and Casclla, G. (1996), "The Effect of Improper Priors on Gibbs Sampling in Hierarchical Linear Mixed Models," Journal of the American Statistical Association, 91, 1461-1474.
16. Hoffman, D. K., Raffenetti, R. C., and Ruedenberg, K. (1972), "Generalization of Euler Angles to W-Dimensional Orthogonal Matrices," Journal of Mathematical Physics, 13, 528-533.
17. Kass, R. E., and Wasserman, L. (1996), "The Selection of Prior Distributions by Formal Rules," Journal of the American Statistical Association, 91, 1343-1370.
18. Ledoit, O. (1996). "A Well-Conditioned Estimator for Large Dimensional Covariance Matrices," Working paper. University of Californ:a-Los Angeles. Anderson Graduate School of Management.
19. Leonard, T., and Hsu, J. S. (1993), "Bayesian Inference for a Covariance Matrix," The Annals of Statistics, 21, 1-25.
20. Lin, S. P., and Perlman, M. D. (1985), "A Monte Carlo Comparison of Four Estimators for a Covariance Matrix," in Multivariate Analysis 6, ed. P. R. Krishnaiah, Amsterdam: North-Holland, pp. 411-429.
21. Natarajan, R., and McCulloch, C. E. (1995), "A Note on the Existence of the Posterior Distribution for a Class of Mixed Models for Binomial Responses," Biometrika, 82, 639-643.
22. O'Hagan, A. (1994), Kendall's Advanced Theory of Statistics, Volume IB-Bay esian Inference, New York: Halsted Press.
23. Pinheiro, J. C., and Bates, D. M. (1996), "Unconstrained Paramcterizations for Variance-Covariance Matrices." Statistics and Computing, 6. 1-6.
24. Schervish, M. J. (1995), Theory of Statistics, New York: Springer-Verlag.
25. Smith, A. F. M., and Roberts, G. O. (1993), "Bayesian Compulation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods," Journal of the Royal Statistical Society, Ser. B. 55, 3-23.
26. Stein, C. (1975), "Estimation of a Covariance Matrix," Rietz Lecture, 39th Annual meeting IMS. Atlanta. Georgia.
27. Sun, L., Hsu, J. S., Guttman, I., Leonard, T. (1996). "Bayesian Methods for Variance Component Models." Journal of the American Statistical Association, 91, 743-752.
28. Tanner, M. A. (1993), Tools for Statistical Inference: Methods for Expo-ration of Posterior Distributions and Likelihood Functions, New York: Springer-Verlag.
29. Thisted, R. A. (1988), Elements of Statistical Computing, London: Chapman and Hall.
30. Yang, R., and Berger, J. O. (1994), "Estimation of a Covariance Matrix Using the Reference Prior," The Annals of Statistics, 22. 1195-1211.