Main Article Content
In this paper, we are interested in estimating a multivariate normal mean under the balanced loss function using the shrinkage estimators deduced from the Maximum Likelihood Estimator (MLE). First, we consider a class of estimators containing the James-Stein estimator, we then show that any estimator of this class dominates the MLE, consequently it is minimax. Secondly, we deal with shrinkage estimators which are not only minimax but also dominate the James- Stein estimator.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
- Amin, M., Nauman, A. M., and Amanullah, M. (2020). On the james-stein estimator for the poisson regression model. Comm. Statist. Simulation Comput., pages 1–14, doi.org/10.1080/03610918.2020.1775851. DOI: https://doi.org/10.1080/03610918.2020.1775851
- Arnold, F. S. (1981). The theory of Linear models and Multivariate analysis, pp. 9–10. John Wiley & Sons, New York, USA.
- Benkhaled, A. and Hamdaoui, A. (2019). General classes of shrinkage estimators for the multivariate normal mean with unknown variancee: Minimaxity and limit of risks ratios. Kragujevac J. Math., 46:193–213.
- Guikai, H., Qingguo, L., and Shenghua, L. (2014). Trisk comparison of improved estimators in a linear regression model with multivariate t errors under balanced loss function. Journal of Applied Mathematics, 354:1–7, doi.org/10.1155/2014/129205. DOI: https://doi.org/10.1155/2014/129205
- Hamdaoui, A., Benkhaled, A., and Mezouar, N. (2020). Minimaxity and limits of risks ratios of shrink- age estimators of a multivariate normal mean in the bayesian case. Stat. Optim. Inf. Comput., 8:507–520, doi.org/10.19139/soic–2310–5070–735. DOI: https://doi.org/10.19139/soic-2310-5070-735
- Jafari, J., Leblan, A., and Marchand, E. (2014). On continuous distribution functions, minimax and bes invariant estimators and integrated balanced loss functions. Canad. J. Statistist., 42:470–486, doi.org/10.1002/cjs.11217. DOI: https://doi.org/10.1002/cjs.11217
- James, W. and Stein, C. (1961). Estimation with quadratic loss. Fourth Berkeley Symp. on Math. Statist. and Prob., Univ. of Calif. Press, Berkeley, 1:361–379.
- Karamikabir, H., Afsahri, M., and Arashi, M. (2018). Shrinkage estimation of non-negative mean vec- tor with unknown covariance under balance loss. Journal of Inequalities and Applications, pages 1–11, doi.org/10.1186/s13660–018–1919–0. DOI: https://doi.org/10.1186/s13660-018-1919-0
- Sanjari, F. and Asgharzadeh, A. (2004). Estimation of a normal mean relative to balanced loss functions. Statistical Papers, 45:279–286, doi.org/10.1007/BF02777228. DOI: https://doi.org/10.1007/BF02777228
- Selahattin, K. and Issam, D. (2019). The optimal extended balanced loss function estimators. J. Comput. Appl. Math., 345:86–98, doi.org/10.1016/j.cam.2018.06.021. DOI: https://doi.org/10.1016/j.cam.2018.06.021
- Selahattin, K., Sadullah, S., ozkale, M. R., and G ¨ uler, H. (2011). More on the restricted ridge regression ¨ estimation. J. Stat. Comput. Simul., 81:1433–1448, doi.org/10.1080/00949655.2010.491480. DOI: https://doi.org/10.1080/00949655.2010.491480
- Stein, C. (1956). Inadmissibilty of the usual estimator for the mean of a multivariate normal distribution. Proc. 3th Berkeley Symp. on Math. Statist. and Prob., Univ. of Calif. Press, Berkeley, 1:197–206. DOI: https://doi.org/10.1525/9780520313880-018
- Stein, C. (1981). Estimation of the mean of a multivariate normal distribution. International Statistical Review, 9:1135–1151, doi.org/10.1214/aos/1176345632. DOI: https://doi.org/10.1214/aos/1176345632
- Tsukuma, H. and Kubukawa, T. (2015). Estimation of the mean vector in a singular multivariate normal distribution. J. Multivariate Anal., 140:215–232, doi.org/10.1016/j.jmva.2015.05.016. DOI: https://doi.org/10.1016/j.jmva.2015.05.016
- Xie, X., Kou, S. C., and Brown, B. (2016). Optimal shrinkage estimators of mean parameters in family of distribution with quadratic variance. Ann. Statist., 44:564–597, doi.org/10.1214/15–AOS1377. DOI: https://doi.org/10.1214/15-AOS1377
- Yuzba, B., Arashi, M., and Ahmed, S. (2020). Shrinkage estimation strategies in generalised ridge regression models: Low/high-dimension regime. pages 1–23, doi.org/10.1111/insr.12351. Int. Stat. Rev. DOI: https://doi.org/10.1111/insr.12351
- Zellner, A. (1994). Bayesian and non-bayesian estimation using balanced loss functions. Statistical Decision Theory and Related Topics, 8:337–390. DOI: https://doi.org/10.1007/978-1-4612-2618-5_28
- Zinodiny, S., Leblan, S., and Nadarajah, S. (2017). Bayes minimax estimation of the mean matrix of matrix-variate normal distribution under balanced loss function. Statist. Probab. Lett., 125:110–120, doi.org/10.1016/j.spl.2017.02.003. DOI: https://doi.org/10.1016/j.spl.2017.02.003