### Abstract

In this article we investigate the estimation problem of the population mean of a finite population. Both point and interval estimators are of interest from Bayes and empirical Bayes point of views. Empirical Bayes analysis is concerned with the 'current' population mean, say γ_{m}, when the sample data are available from other similar (m-1) finite populations, Y_{1},..., Y_{m-1}, as well as the data from the current population, Y_{m}, where Y_{i} = (Y_{i1},..., Y_{ini}), i = 1,..., m. Previous results on inference of γ_{m} have assumed either a normal model or a posterior linearity condition in making Bayes inference which is the kernel of the empirical Bayes problem. They resulted in examination of linear estimators of the sample mean Ȳ_{m} = n_{m}^{-1} ∑_{j=1}^{n}m Y_{mj}. In this paper, we propose to investigate a generalizing idea which generates optimal linear Bayes estimators of γ_{m} as functions of Ȳ_{m}. We develop optimal linear Bayes estimators of γ_{m} under two Bayesian models. They are optimal in the sense of minimizing the mean squared error with respect to the underlying models. The corresponding empirical Bayes analogues are obtained by replacing the unknown hyperparameters by their respective consistent estimates as usual. The asymptotic optimality criterion is employed in order to measure the goodness of the proposed empirical Bayes estimators. Very promising Bayes and empirical Bayes two-sided confidence intervals and predictors of γ_{m} are also discussed. A Monte Carlo study is conducted to evaluate the performance of the proposed estimators.

Original language | English (US) |
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Pages (from-to) | 505-525 |

Number of pages | 21 |

Journal | Journal of Statistical Planning and Inference |

Volume | 113 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2003 |

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### Keywords

- Asymptotic optimality
- Bayes estimators
- Empirical Bayes
- Finite populations
- Linear estimators

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics

### Cite this

*Journal of Statistical Planning and Inference*,

*113*(2), 505-525. https://doi.org/10.1016/S0167-7152(01)00206-1

**Optimal linear Bayes and empirical Bayes estimation and prediction of the finite population mean.** / Karunamuni, Rohana J.; Zhang, Shunpu.

Research output: Contribution to journal › Article

*Journal of Statistical Planning and Inference*, vol. 113, no. 2, pp. 505-525. https://doi.org/10.1016/S0167-7152(01)00206-1

}

TY - JOUR

T1 - Optimal linear Bayes and empirical Bayes estimation and prediction of the finite population mean

AU - Karunamuni, Rohana J.

AU - Zhang, Shunpu

PY - 2003/5/1

Y1 - 2003/5/1

N2 - In this article we investigate the estimation problem of the population mean of a finite population. Both point and interval estimators are of interest from Bayes and empirical Bayes point of views. Empirical Bayes analysis is concerned with the 'current' population mean, say γm, when the sample data are available from other similar (m-1) finite populations, Y1,..., Ym-1, as well as the data from the current population, Ym, where Yi = (Yi1,..., Yini), i = 1,..., m. Previous results on inference of γm have assumed either a normal model or a posterior linearity condition in making Bayes inference which is the kernel of the empirical Bayes problem. They resulted in examination of linear estimators of the sample mean Ȳm = nm-1 ∑j=1nm Ymj. In this paper, we propose to investigate a generalizing idea which generates optimal linear Bayes estimators of γm as functions of Ȳm. We develop optimal linear Bayes estimators of γm under two Bayesian models. They are optimal in the sense of minimizing the mean squared error with respect to the underlying models. The corresponding empirical Bayes analogues are obtained by replacing the unknown hyperparameters by their respective consistent estimates as usual. The asymptotic optimality criterion is employed in order to measure the goodness of the proposed empirical Bayes estimators. Very promising Bayes and empirical Bayes two-sided confidence intervals and predictors of γm are also discussed. A Monte Carlo study is conducted to evaluate the performance of the proposed estimators.

AB - In this article we investigate the estimation problem of the population mean of a finite population. Both point and interval estimators are of interest from Bayes and empirical Bayes point of views. Empirical Bayes analysis is concerned with the 'current' population mean, say γm, when the sample data are available from other similar (m-1) finite populations, Y1,..., Ym-1, as well as the data from the current population, Ym, where Yi = (Yi1,..., Yini), i = 1,..., m. Previous results on inference of γm have assumed either a normal model or a posterior linearity condition in making Bayes inference which is the kernel of the empirical Bayes problem. They resulted in examination of linear estimators of the sample mean Ȳm = nm-1 ∑j=1nm Ymj. In this paper, we propose to investigate a generalizing idea which generates optimal linear Bayes estimators of γm as functions of Ȳm. We develop optimal linear Bayes estimators of γm under two Bayesian models. They are optimal in the sense of minimizing the mean squared error with respect to the underlying models. The corresponding empirical Bayes analogues are obtained by replacing the unknown hyperparameters by their respective consistent estimates as usual. The asymptotic optimality criterion is employed in order to measure the goodness of the proposed empirical Bayes estimators. Very promising Bayes and empirical Bayes two-sided confidence intervals and predictors of γm are also discussed. A Monte Carlo study is conducted to evaluate the performance of the proposed estimators.

KW - Asymptotic optimality

KW - Bayes estimators

KW - Empirical Bayes

KW - Finite populations

KW - Linear estimators

UR - http://www.scopus.com/inward/record.url?scp=0037408753&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037408753&partnerID=8YFLogxK

U2 - 10.1016/S0167-7152(01)00206-1

DO - 10.1016/S0167-7152(01)00206-1

M3 - Article

AN - SCOPUS:0037408753

VL - 113

SP - 505

EP - 525

JO - Journal of Statistical Planning and Inference

JF - Journal of Statistical Planning and Inference

SN - 0378-3758

IS - 2

ER -