A Class of Estimators Dominating the Lindley Type Estimator Under the Balanced Loss Function

doi:10.22716/sckt.2023.11.3.023

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2023 Vol.11, Issue 3 Preview Page Next Page

Research Article

A Class of Estimators Dominating the Lindley Type Estimator Under the Balanced Loss Function 균형 잡힌 손실 함수 하에서 Lindley 형 추정량을 개량하는 추정량들의 일련의 유형: In-Bong Choi¹, Jeong-Mi Lee², Hoh-Yoo Baek³
최인봉¹, 이정미², 백호유³; ¹Research Commissioner, Korea Institute For Curriculum and Evaluation
²Associate Professor, Department of Public Health, Wonkwang University Graduate School
³Professor Emeritus, Division of Big Data and Financial Statistics, Wonkwang University

¹한국교육과정평가원 연구위원
²원광대학교 일반대학원 보건학과 부교수
³원광대학교 빅데이터금융통계학부 명예교수

30 September 2023. pp. 27-43

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Abstract

본 논문은 균형 잡힌 손실 함수의 기댓값인 위험함수를 비교하여 다변량 정규 분포함수에서 평균의 수축형 추정량들과 비교되는 개량 관계와 최대 최소성을 다루었다. 본 연구에서는 두 가지 서로 다른 추정량의 유형들을 제시하였다. 첫째로, Lindley 형 추정량을 일반화하고, 임의의 추정량이 최대가능도 추정량을 개량함을 보인다. 결과적으로 그것은 최대최소의 성질을 만족한다. 둘째로, 그 추정량이 Lindley 형 추정량 개량함을 위험함수의 비교를 통하여 확인하였고, 이러한 유형의 추정량 또한 최대 최소성을 갖는다는 결론을 내릴 수 있다. 아울러, 제시된 여러 추정량들의 실행 예시에 따른 모의실험 연구 결과를 표와 그림으로 제시하였다.

In this paper we are dealing with the shrinkage estimators of a multivariate normal mean and their minimaxity properties under the balanced loss function by comparing risk function of the loss. This paper is presented here two different classes of estimator. First, we generalizes the Lindley type estimator and show that any estimator of this class dominates the maximum likelihood estimator (MLE), consequently it is minimax. Second, we can also show that it dominates the Lindley type and conclude that any estimator of this class is minimax. In addtion to, we conduct a simulation study that shows the performance of the considered estimators by using various tables and figures.

Keywords

Balanced loss function

Lindley type estimator

Minimax estimator

Multivariate normal mean

Shrinkage estimator

References

C. Stein, "Inadmissibility 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. 10.1525/9780520313880-018
W. James, and C. Stein, "Estimation with quadratic loss", Proc 4th Berkeley Symp, Math. Statist.Prob., Univ of California Press, Berkeley, Vol. 1, pp. 361-379, 1961.
R. Yang, and J.O. Berger, "Estimation of a covariance matrix using the reference prior", Ann. Statist., Vol. 22, No. 3, pp. 1195-1211, 1994. 10.1214/aos/1176325625
B. Efron, and C.N. Morris, "Stein's estimation rule and its competitors: An empirical Bayes approach", J. Amer. Statist. Assoc., Vol. 68, pp. 117-130, 1973. 10.1080/01621459.1973.10481350
B. Efron, and C.N. Morris, "Data analysis using Stein's estimator and its generalizations", J. Amer. Statist. Assoc., Vol. 70, pp. 311-319, 1975. 10.1080/01621459.1975.10479864
L.D. Brown, "In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies", Ann. Appl. Stat., Vol. 2, No. 1, pp. 113-152, 2008. 10.1214/07-AOAS138
A. Benkhaled, and A. Hamdaoui, "General classes of shrinkage estimators for the multivariate normal mean with unknown variancee: Minimaxity and limit of risks ratios", Kragujevac J. Math., Vol. 46, No. 2, pp. 193-213, 2019. 10.46793/KgJMat2202.193B
D.V. Lindley, "Discussion of paper by C. Stein", J. Roy. Statist. Soc. Ser. B., Vol. 24, pp. 265-596, 1962.
A. Zellner, "Bayesian and non-Bayesian estimation using balanced loss functions", In: Berger, J.O., Gupta, S.S. (eds.) Statistical Decision Theory and Methods, Vol. V, pp. 337-390, Springer, New York, NY. 1994. 10.1007/978-1-4612-2618-5_28
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. 10.1155/2014/129205
H. Karamikabir, M. Afsahri, and M. Arashi, "Shrinkage estimation of non-negative mean vector with unknown covariance under balance loss", J. Inequal. Appl., Vol. 2018, No. 1, pp. 1-11, 2018. 10.1186/s13660-018-1919-0 30839820 PMC6280813
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. 10.1016/j.spl.2017.02.003
F. Arnold Steven, "The theory of linear models and multivariate analysis", John Wiley and Sons, Inc., pp. 9-10, 1981.
C. Stein, "Estimation of the mean of a multivariate normal distribution", Ann. Statist., Vol. 9, No. 6, pp. 1135-1151, 1981. 10.1214/aos/1176345632
M. Jafari Jozani, A. Leblanc, and E. Marchand, "On continuous distribution functions, minimax and best invariant estimators and integrated balanced loss functions", Canad. J. Statistist., Vol. 42, No. 3, pp. 470-486, 2014. 10.1002/cjs.11217
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. 10.1016/j.jmva.2019.104583
A. Benkhaled, A. Hamdaoui, and M. Terbeche, "Minimax shrinkage estimators and estimators dominating the James-Stein estimator under the balanced loss function", Eurasian Math. J., Vol. 13, No. 2. pp. 18-36, 2022. 10.32523/2077-9879-2022-13-2-18-36

Information

Publisher :The Society of Convergence Knowledge
Publisher(Ko) :융복합지식학회
Journal Title :The Society of Convergence Knowledge Transactions
Journal Title(Ko) :융복합지식학회논문지
Volume : 11
No :3
Pages :27-43
DOI :https://doi.org/10.22716/sckt.2023.11.3.023

[1] C. Stein, "Inadmissibility 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. 10.1525/9780520313880-018

[2] W. James, and C. Stein, "Estimation with quadratic loss", Proc 4th Berkeley Symp, Math. Statist.Prob., Univ of California Press, Berkeley, Vol. 1, pp. 361-379, 1961.

[3] R. Yang, and J.O. Berger, "Estimation of a covariance matrix using the reference prior", Ann. Statist., Vol. 22, No. 3, pp. 1195-1211, 1994. 10.1214/aos/1176325625

[4] B. Efron, and C.N. Morris, "Stein's estimation rule and its competitors: An empirical Bayes approach", J. Amer. Statist. Assoc., Vol. 68, pp. 117-130, 1973. 10.1080/01621459.1973.10481350

[5] B. Efron, and C.N. Morris, "Data analysis using Stein's estimator and its generalizations", J. Amer. Statist. Assoc., Vol. 70, pp. 311-319, 1975. 10.1080/01621459.1975.10479864

[6] L.D. Brown, "In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies", Ann. Appl. Stat., Vol. 2, No. 1, pp. 113-152, 2008. 10.1214/07-AOAS138

[7] A. Benkhaled, and A. Hamdaoui, "General classes of shrinkage estimators for the multivariate normal mean with unknown variancee: Minimaxity and limit of risks ratios", Kragujevac J. Math., Vol. 46, No. 2, pp. 193-213, 2019. 10.46793/KgJMat2202.193B

[8] D.V. Lindley, "Discussion of paper by C. Stein", J. Roy. Statist. Soc. Ser. B., Vol. 24, pp. 265-596, 1962.

[9] A. Zellner, "Bayesian and non-Bayesian estimation using balanced loss functions", In: Berger, J.O., Gupta, S.S. (eds.) Statistical Decision Theory and Methods, Vol. V, pp. 337-390, Springer, New York, NY. 1994. 10.1007/978-1-4612-2618-5_28

[10] 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. 10.1155/2014/129205

[11] H. Karamikabir, M. Afsahri, and M. Arashi, "Shrinkage estimation of non-negative mean vector with unknown covariance under balance loss", J. Inequal. Appl., Vol. 2018, No. 1, pp. 1-11, 2018. 10.1186/s13660-018-1919-0 30839820 PMC6280813

[12] 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. 10.1016/j.spl.2017.02.003

[13] F. Arnold Steven, "The theory of linear models and multivariate analysis", John Wiley and Sons, Inc., pp. 9-10, 1981.

[14] C. Stein, "Estimation of the mean of a multivariate normal distribution", Ann. Statist., Vol. 9, No. 6, pp. 1135-1151, 1981. 10.1214/aos/1176345632

[15] M. Jafari Jozani, A. Leblanc, and E. Marchand, "On continuous distribution functions, minimax and best invariant estimators and integrated balanced loss functions", Canad. J. Statistist., Vol. 42, No. 3, pp. 470-486, 2014. 10.1002/cjs.11217

[16] 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. 10.1016/j.jmva.2019.104583

[17] A. Benkhaled, A. Hamdaoui, and M. Terbeche, "Minimax shrinkage estimators and estimators dominating the James-Stein estimator under the balanced loss function", Eurasian Math. J., Vol. 13, No. 2. pp. 18-36, 2022. 10.32523/2077-9879-2022-13-2-18-36

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