DOI QR코드

DOI QR Code

How to Improve Classical Estimators via Linear Bayes Method?

  • Wang, Lichun (Department of Mathematics, Beijing Jiaotong University)
  • 투고 : 2015.10.17
  • 심사 : 2015.10.26
  • 발행 : 2015.11.30

초록

In this survey, we use the normal linear model to demonstrate the use of the linear Bayes method. The superiorities of linear Bayes estimator (LBE) over the classical UMVUE and MLE are established in terms of the mean squared error matrix (MSEM) criterion. Compared with the usual Bayes estimator (obtained by the MCMC method) the proposed LBE is simple and easy to use with numerical results presented to illustrate its performance. We also examine the applications of linear Bayes method to some other distributions including two-parameter exponential family, uniform distribution and inverse Gaussian distribution, and finally make some remarks.

키워드

참고문헌

  1. Arnold, S. F. (1980). The Theory of Linear Models and Multivariate Analysis, John Wiley & Sons, New York.
  2. Busby, D., Farmer, C. L. and Iske, A. (2005). Uncertainty evaluation in reservoir forecasting by Bayes linear methodology, Algorithms for approximation, proceedings of the 5th international conference, Chester, 187-196.
  3. Casella, G. and George, E. I. (1992). An introduction to Gibbs Sampling, The American Statistician, 46, 167-174.
  4. Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities, Journal of the American Statistical Association, 85, 398-409. https://doi.org/10.1080/01621459.1990.10476213
  5. Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American Statistical Association, 85, 972-985. https://doi.org/10.1080/01621459.1990.10474968
  6. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Transactions PAMI, 6, 721-741.
  7. Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo in Practice, Chapman and Hall, London.
  8. Goldstein, M. (1983). General variance modifications for linear Bayes estimators, Journal of the American Statistical Association, 78, 616-618. https://doi.org/10.1080/01621459.1983.10478019
  9. Hartigan, J. A. (1969). Linear Bayesian methods, Journal of the Royal Statistical Society, Series B, 31, 440-454.
  10. Heiligers, B. (1993). Linear Bayes and minimax estimation in linear models with partially restricted parameter space, Journal of Statistical Planning and Inference, 36, 175-183. https://doi.org/10.1016/0378-3758(93)90122-M
  11. Hoffmann, K. (1996). A subclass of Bayes linear estimators that are minimax, Acta Applicandue Mathematicae, 43, 87-95. https://doi.org/10.1007/BF00046990
  12. Lamotte, L. R. (1978). Bayes linear estimators, Technometrics, 3, 281-290.
  13. Lindley, D. V. (1980). Approximate Bayesian methods, Trabajos de Estadistica, 21, 223-237.
  14. Mao, S. S. and Tang, Y. C. (2012). Bayesian Statistics, Second Edition, China Statistics Press, Beijing.
  15. Martinez, W. L. and Martinez, A. R. (2007). Computational Statistics Handbook with MATLAB, Second Edition, Chapman & Hall/CRC, New York.
  16. Pensky, M. and Ni, P. (2000). Extended linear empirical Bayes estimation, Communications in Statistics - Theory and Methods, 29, 579-592. https://doi.org/10.1080/03610920008832503
  17. Rao, C. R. (1973). Linear Statistical Inference and Its Applications, Second Edition, John Wiley & Sons, New York.
  18. Robert, C. P. and Casella, G.(1999). Monte Carlo Statistical Methods, Springer-Verlag, New York.
  19. Samaniego, F. J. and Vestrup, E. (1999). On improving standard estimators via linear empirical Bayes methods, Statistics & Probability Letters, 44, 309-318. https://doi.org/10.1016/S0167-7152(99)00022-X
  20. Schwarz, C. J. and Samanta, M. (1991). An inductive proof of the sampling distributions for the MLEs of the parameters in an inverse Gaussian distributions, The American Statistician, 45, 223-235.
  21. Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities, Journal of the American Statistical Association, 81, 82-86. https://doi.org/10.1080/01621459.1986.10478240
  22. Tweedie, M. C. K. (1957). Statistical properties of inverse Gaussian distributions, The Annals of Mathematical Statistics, 28, 362-377. https://doi.org/10.1214/aoms/1177706964
  23. Wang, L. C. and Singh, R. S. (2014). Linear Bayes estimator for the two-parameter exponential family under type II censoring, Computational Statistics and Data Analysis, 71, 633-642. https://doi.org/10.1016/j.csda.2013.07.020
  24. Wei, L. S. and Zhang, W. P. (2007). The superiorities of Bayes linear minimum risk estimation in linear model, Communications in Statistics-Theory and Methods, 36, 917-926. https://doi.org/10.1080/03610920601036333
  25. Zhang, W. P. and Wei, L. S. (2005). On Bayes linear unbiased estimation of estimable functions for the singular linear model, Science in China Series A Mathematics, 7, 898-903.
  26. Zhang, W. P., Wei, L. S. and Chen, Y. (2011). The superiorities of Bayes linear unbiased estimation in partitioned linear model, Journal of System Science and Complexity, 5, 945-954.
  27. Zhang, W. P., Wei, L. S. and Chen, Y. (2012). The superiorities of Bayes linear unbiased estimation in multivariate linear models, Acta Mathematicae Applicatae Sinica, English Series 2, 383-394.