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Shrinkage Testimator for the Common Mean of Several Univariate Normal Populations

Author

Listed:
  • Peter M. Mphekgwana

    (Department of Statistics, Faculty of Science and Agriculture, Fort Hare University, Alice 5700, South Africa
    Department of Research Administration and Development, University of Limpopo, Sovenga 0727, South Africa)

  • Yehenew G. Kifle

    (Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA)

  • Chioneso S. Marange

    (Department of Statistics, Faculty of Science and Agriculture, Fort Hare University, Alice 5700, South Africa)

Abstract

The challenge of combining two unbiased estimators is a common occurrence in applied statistics, with significant implications across diverse fields such as manufacturing quality control, medical research, and the social sciences. Despite the widespread relevance of estimating the common population mean μ , this task is not without its challenges. A particularly intricate issue arises when the variations within populations are unknown or possibly unequal. Conventional approaches, like the two-sample t -test, fall short in addressing this problem as they assume equal variances among the two populations. When there exists prior information regarding population variances ( σ i 2 , i = 1 , 2 ) , with the consideration that σ 1 2 and σ 2 2 might be equal, a hypothesis test can be conducted: H 0 : σ 1 2 = σ 2 2 versus H 1 : σ 1 2 ≠ σ 2 2 . The initial sample is utilized to test H 0 , and if we fail to reject H 0 , we gain confidence in incorporating our prior knowledge (after testing) to estimate the common mean μ . However, if H 0 is rejected, indicating unequal population variances, the prior knowledge is discarded. In such cases, a second sample is obtained to compensate for the loss of prior knowledge. The estimation of the common mean μ is then carried out using either the Graybill–Deal estimator (GDE) or the maximum likelihood estimator (MLE). A noteworthy discovery is that the proposed preliminary testimators, denoted as μ ^ P T 1 and μ ^ P T 2 , exhibit superior performance compared to the widely used unbiased estimators (GDE and MLE).

Suggested Citation

  • Peter M. Mphekgwana & Yehenew G. Kifle & Chioneso S. Marange, 2024. "Shrinkage Testimator for the Common Mean of Several Univariate Normal Populations," Mathematics, MDPI, vol. 12(7), pages 1-18, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1095-:d:1370687
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    References listed on IDEAS

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    1. Pal, Nabendu & Lin, Jyh-Jiuan & Chang, Ching-Hui & Kumar, Somesh, 2007. "A revisit to the common mean problem: Comparing the maximum likelihood estimator with the Graybill-Deal estimator," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5673-5681, August.
    2. Giles, Judith A & Giles, David E A, 1993. "Pre-test Estimation and Testing in Econometrics: Recent Developments," Journal of Economic Surveys, Wiley Blackwell, vol. 7(2), pages 145-197, June.
    3. Philip Yu & Yijun Sun & Bimal Sinha, 2002. "Estimation of the Common Mean of a Bivariate Normal Population," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(4), pages 861-878, December.
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