Polynomials Shrinkage Estimators of a Multivariate Normal Mean

  • Abdelkader Benkhaled Mascara University, Algeria
  • Mekki Terbeche Algeria
  • Abdenour Hamdaoui University of Sciences and Technologies Mohamed Boudiaf, Oran (USTO).Department of Mathematics. Algeria
DOI: https://doi.org/10.19139/soic-2310-5070-1095
Keywords: Balanced Loss Function, James-Stein estimator, multivariate Gaussian random variable, non-central chi-square distribution, shrinkage estimators.

Abstract

In this work, the estimation of the multivariate normal mean by different classes of shrinkage estimators is investigated. The risk associated with the balanced loss function is used to compare two estimators. We start by considering estimators that generalize the James-Stein estimator and show that these estimators dominate the maximum likelihood estimator (MLE), therefore are minimax, when the shrinkage function satisfifies some conditions. Then, we treat estimators of polynomial form and prove the increase of the degree of the polynomial allows us to build a better estimator from the one previously constructed.

Author Biography

Abdenour Hamdaoui, University of Sciences and Technologies Mohamed Boudiaf, Oran (USTO).Department of Mathematics. Algeria
Doctorat in Mathematics (PhD) option of Probability and statistics at the Department of Mathematics, University of Tlemcen, Algeria. Senior lecturer at the Department of Mathematics, University of Science and Technology of Oran Mohamed Boudiaf (USTO-MB), Oran, Algeria.

References

M. Amin, M. Nauman Akram, and M. Amanullah, On the James-Stein estimator for the Poisson regression model, Comm. Statist. Simulation Comput., pp. 1–14, 2020.

F. Arnold Steven, The theory of linear models and multivariate analysis, John Wiley and Sons, Inc., pp. 9–10, 1981.

A. Benkhaled, and A. Hamdaoui, General classes of shrinkage estimators for the multivariate normal mean with unknown variance: Minimaxity and limit of risks ratios, Kragujevac J. Math., vol. 46, pp. 193–213, 2019.

H. Guikai, L. Qingguo, and Y. Shenghua, Risk Comparison of improved estimators in a linear regression model with multivariate errors under Balanced Loss Function, Journal of Applied Mathematics, vol. 354, pp. 1–7, 2014.

A. Hamdaoui, A. Benkhaled, and N. Mezouar, Minimaxity and limits of risks ratios of shrinkage estimators of a multivariate normal mean in the Bayesian case, Stat., Optim. Inf. Comput., vol. 8(2), pp. 507–520, 2020.

A. Hamdaoui, A. Benkhaled, and M. Terbeche, Baranchick-type estimators of a multivariate normal mean under the general quadratic loss function, Journal of Siberian Federal University. Mathematics and Physics., vol. 13(5), pp. 608–621, 2020.

M. Jafari Jozani, A. Leblan, and E. Marchand, On continuous distribution functions, minimax and best invariant estimators and integrated balanced loss functions, Canad. J. Statist., vol. 42, pp. 470–486, 2014.

W. James, and C. Stein, Estimation of quadratique loss, Proc 4th Berkeley Symp, Math. Statist.Prob, Univ of California Press, Berkeley., vol. 1 , pp. 361-379, 1961.

H. Karamikabir, and M. Afsahri, Generalized Bayesian shrinkage and wavelet estimation of location parameter for spherical distribution under balanced-type loss: Minimaxity and admissibility, J. Multivariate Anal., vol. 177, pp. 110–120, 2020.

H. Karamikabir, M. Afshari, and M. Arashi, Shrinkage estimation of non-negative mean vector with unknown covariance under balance loss, Journal of Inequalities and Applications, pp, 1–11, 2018.

N. Sanjari Farsipour, and A. Asgharzadeh, Estimation of a normal mean relative to balanced loss functions, Statistical Papers, vol. 45, pp. 279–286, 2004.

K. Selahattin, and D. Issam, The optimal extended balanced loss function estimators, J. Comput. Appl. Math., vol. 345, pp. 86-98, 2019.

C. Stein, Estimation of the mean of a multivariate normal distribution, Ann. Statis., vol. 9(6), pp. 1135–1151, 1981.

C. Stein, 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, vol. 1, pp. 197–206, 1956.

B. Yuzba, M. Arashi, and S. Ejaz, Ahmed, Shrinkage estimation strategies in generalized ridge regression models: Low/High-dimension regime, Int. Stat. Rev., pp. 1–23, 2020.

A. Zellner, Bayesian and non-Bayesian estimation using balanced loss functions, In: Berger, J.O., Gupta, S.S. (eds.) Statistical Decision Theory and Methods, Volume V, pp. 337–390. Springer, New York 1994.

S. Zinodiny, S. Leblan, and S. Nadarajah, Bayes minimax estimation of the mean matrix of matrix-variate normal distribution under balanced loss function, Statist. Probab. Lett., vol. 125, pp. 110–120, 2017.

Published
2021-08-17
How to Cite
Benkhaled, A., Terbeche, M., & Hamdaoui, A. (2021). Polynomials Shrinkage Estimators of a Multivariate Normal Mean. Statistics, Optimization & Information Computing, 10(2), 572-588. https://doi.org/10.19139/soic-2310-5070-1095
Issue
Vol 10 No 2 (2022)
Section
Research Articles
Authors who publish with this journal agree to the following terms:

    1. 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.
    2. 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.
    3. 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).