Keywords:-
Keywords:
Modal Bayes estimator, multivariate normal distribution, quadratic risk, shrinkage estimator.
Article Content:-
Abstract
This article concerns the estimation of the mean of a multivariate normal distribution in which the variance is unknown and estimated by the chi-square variable . First, we consider the estimators of Lindley-Type that shrink the components of the Maximum Likelihood Estimator (MLE) to the random variable . Secondly, we consider the mean as a random variable and construct the modal Bayes estimator , we then study the minimaxity of the estimator and the asymptotic behavior of risks ratios of to the MLE when the dimension of the parameters space and the sample size tend simultaneously to infinity.
References:-
References
1. Baranchik, A. J. (1964). Multiple regression and estimation of the mean of a multivariate normal distribution. Stanford Univ. Technical Report, 51.
2. Baranchick, A. J. (1970). A family of minimax estimators of the mean of a multivariate normal distribution, The Annals of Mathematical Statistics, 41(2), 642–645.
3. Benkhaled, A., and Hamdaoui, A. (2019). General classes of shrinkage estimators for the
4. multivariate normal mean with unknown variance: minimaxity and limit of risks ratios, Kragujevac J.Math., 46(2), 193–213.
5. Benmansour, D., and Hamdaoui. A. (2011). Limit of the Ratio of Risks of James-Stein Estimators with Unknown Variance. Far East Journal of Theorical Statistics; Vol. 36(1), 31-53.
6. Bock, M.E. (1975). Minimax estimators of the mean of a multivariate normal distribution, Ann.Statist., 3(1), 209–218.
7. Brandwein, A.C. , and Strawderman, W. E. (2012). Stein Estimation for Spherically
8. Symmetric Distributions : Recent Developments. Statistical Science, 27(1), 11-23.
9. Casella, G., and Hwang, J.T. (1982). Limit expressions for the risk of James-Stein estimators. The canadian Journal of Statistics. 10(4), 305-309.
10. Efron, B., and Morris, C. (1972). Limiting the Risk of Bayes and Empirical Bayes Estimators – Part II: The Empirical Bayes Case. Journal of the American Statistical Association.67,130-139.
11. Efron, B., and Morris, C. (1973). Stein’s Estimation Rule and Its Competitors – An Empirical Bayes Approach. Journal of the American Statistical Association, 68, 117-130.
12. Efron, B., and Morris, C., (1977). Stein's Paradox in Statistics. Scientific American. 236 (5): 119–127.
13. Hamdaoui, A., and Benmansour. D. (2015). Asymptotic properties of risks ratios of shrinkage estimators. Hacettepe Journal of Mathematics and Statistics. 44(5), 1181 – 1195.
14. Hamdaoui, A., Benkhaled. D., and Mezouar. N. (2020). Minimaxity and Limits of Risks Ratios of Shrinkage Estimators of a Multivariate Normal Mean in the Bayesian Case. Statistics Optimization and Information Computing. 8(2), 507 – 520.
15. James, W., and Stein. C. (1961). Estimation of quadratique loss. Proc 4th Berkeley Symp, Math .Statist.Prob., Univ. of California Press, Berkeley, 1, 361-379.
16. Lindley, D. V. (1962). Discussion on Professor Stein's paper. J. R. Stat. Soc. Ser. B Stat. Methodol. 24: 285-287.
17. Sun. L. (1995). Risk Ratio and Minimaxity in Estimating the Multivariate Normal Mean with unknown Variance. Scandinavian Journal of Statistics, 22(1). 105-120.
18. Maruyama. Y. (2014). l_{p}-norm based James-Stein estimation with minimaxity and sparsity. Statistics Theory (math.ST) arXiv: 1402-0302.
19. Stein, C. (1956). Inadmissibilty of the usual estimator for the mean of a multivariate normal distribution. Proc 3th Berkeley Symp, Math.Statist. Prob., Univ. of California Press, Berkeley, 1,197-206.
2. Baranchick, A. J. (1970). A family of minimax estimators of the mean of a multivariate normal distribution, The Annals of Mathematical Statistics, 41(2), 642–645.
3. Benkhaled, A., and Hamdaoui, A. (2019). General classes of shrinkage estimators for the
4. multivariate normal mean with unknown variance: minimaxity and limit of risks ratios, Kragujevac J.Math., 46(2), 193–213.
5. Benmansour, D., and Hamdaoui. A. (2011). Limit of the Ratio of Risks of James-Stein Estimators with Unknown Variance. Far East Journal of Theorical Statistics; Vol. 36(1), 31-53.
6. Bock, M.E. (1975). Minimax estimators of the mean of a multivariate normal distribution, Ann.Statist., 3(1), 209–218.
7. Brandwein, A.C. , and Strawderman, W. E. (2012). Stein Estimation for Spherically
8. Symmetric Distributions : Recent Developments. Statistical Science, 27(1), 11-23.
9. Casella, G., and Hwang, J.T. (1982). Limit expressions for the risk of James-Stein estimators. The canadian Journal of Statistics. 10(4), 305-309.
10. Efron, B., and Morris, C. (1972). Limiting the Risk of Bayes and Empirical Bayes Estimators – Part II: The Empirical Bayes Case. Journal of the American Statistical Association.67,130-139.
11. Efron, B., and Morris, C. (1973). Stein’s Estimation Rule and Its Competitors – An Empirical Bayes Approach. Journal of the American Statistical Association, 68, 117-130.
12. Efron, B., and Morris, C., (1977). Stein's Paradox in Statistics. Scientific American. 236 (5): 119–127.
13. Hamdaoui, A., and Benmansour. D. (2015). Asymptotic properties of risks ratios of shrinkage estimators. Hacettepe Journal of Mathematics and Statistics. 44(5), 1181 – 1195.
14. Hamdaoui, A., Benkhaled. D., and Mezouar. N. (2020). Minimaxity and Limits of Risks Ratios of Shrinkage Estimators of a Multivariate Normal Mean in the Bayesian Case. Statistics Optimization and Information Computing. 8(2), 507 – 520.
15. James, W., and Stein. C. (1961). Estimation of quadratique loss. Proc 4th Berkeley Symp, Math .Statist.Prob., Univ. of California Press, Berkeley, 1, 361-379.
16. Lindley, D. V. (1962). Discussion on Professor Stein's paper. J. R. Stat. Soc. Ser. B Stat. Methodol. 24: 285-287.
17. Sun. L. (1995). Risk Ratio and Minimaxity in Estimating the Multivariate Normal Mean with unknown Variance. Scandinavian Journal of Statistics, 22(1). 105-120.
18. Maruyama. Y. (2014). l_{p}-norm based James-Stein estimation with minimaxity and sparsity. Statistics Theory (math.ST) arXiv: 1402-0302.
19. Stein, C. (1956). Inadmissibilty of the usual estimator for the mean of a multivariate normal distribution. Proc 3th Berkeley Symp, Math.Statist. Prob., Univ. of California Press, Berkeley, 1,197-206.
Downloads
Citation Tools
How to Cite
Hamdaoui, A., Benkhaled, A., & Terbeche, M. (2021). Estimators of Lindley-Type for the Multivariate Normal Mean In the Bayesian Case. International Journal Of Mathematics And Computer Research, 9(03), 2203-2209. Retrieved from https://ijmcr.in/index.php/ijmcr/article/view/308