IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i7p1095-d1370687.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/7/1095/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/7/1095/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Badi H. Baltagi & Peter Egger & Michael Pfaffermayr, 2007. "A Monte Carlo Study for Pure and Pretest Estimators of a Panel Data Model with Spatially Autocorrelated Disturbances," Annals of Economics and Statistics, GENES, issue 87-88, pages 11-38.
    2. Chang, Ching-Hui & Pal, Nabendu, 2008. "Testing on the common mean of several normal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 321-333, December.
    3. Matei Demetrescu & Uwe Hassler & Vladimir Kuzin, 2011. "Pitfalls of post-model-selection testing: experimental quantification," Empirical Economics, Springer, vol. 40(2), pages 359-372, April.
    4. Yannick Hoga, 2022. "Quantifying the data-dredging bias in structural break tests," Statistical Papers, Springer, vol. 63(1), pages 143-155, February.
    5. Danilov, D.L. & Magnus, J.R., 2001. "On the Harm that Pretesting Does," Other publications TiSEM f131c709-4db4-468d-9ae8-9, Tilburg University, School of Economics and Management.
    6. S. K. Sapra, 2003. "Pre-test estimation in Poisson regression model," Applied Economics Letters, Taylor & Francis Journals, vol. 10(9), pages 541-543.
    7. Gupta, Ramesh C. & Li, Xue, 2006. "Statistical inference for the common mean of two log-normal distributions and some applications in reliability," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 3141-3164, July.
    8. Clarke, Judith A., 2008. "On weighted estimation in linear regression in the presence of parameter uncertainty," Economics Letters, Elsevier, vol. 100(1), pages 1-3, July.
    9. Lauren Bin Dong, 2004. "Testing for structural Change in Regression: An Empirical Likelihood Ratio Approach," Econometrics Working Papers 0405, Department of Economics, University of Victoria.
    10. N. Balakrishnan & N. Martín & L. Pardo, 2017. "Empirical phi-divergence test statistics for the difference of means of two populations," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 101(2), pages 199-226, April.
    11. Judith Anne Clarke, 2017. "Model Averaging OLS and 2SLS: An Application of the WALS Procedure," Econometrics Working Papers 1701, Department of Economics, University of Victoria.
    12. Pravash Jena & Manas Ranjan Tripathy & Somesh Kumar, 2023. "Point and Interval Estimation of Powers of Scale Parameters for Two Normal Populations with a Common Mean," Statistical Papers, Springer, vol. 64(5), pages 1775-1804, October.
    13. David E. A. Giles, 2000. "Preliminary-Test and Bayes Estimation of a Location Parameter Under 'Reflected Normal' Loss," Econometrics Working Papers 0004, Department of Economics, University of Victoria.
    14. Magnus, J.R. & Wang, W. & Zhang, Xinyu, 2012. "WALS Prediction," Discussion Paper 2012-043, Tilburg University, Center for Economic Research.
    15. Constantinescu, Mihnea & Lastauskas, Povilas, 2018. "The knotty interplay between credit and housing," The Quarterly Review of Economics and Finance, Elsevier, vol. 70(C), pages 241-266.
    16. Guggenberger, Patrik, 2010. "The impact of a Hausman pretest on the size of a hypothesis test: The panel data case," Journal of Econometrics, Elsevier, vol. 156(2), pages 337-343, June.
    17. Danilov, D.L. & Magnus, J.R., 2002. "Estimation of the Mean of a Univariate Normal Distribution When the Variance is not Known," Other publications TiSEM 002a672b-73b6-4a8b-8901-7, Tilburg University, School of Economics and Management.
    18. Reif, Jiri & Vlcek, Karel, 2002. "Optimal pre-test estimators in regression," Journal of Econometrics, Elsevier, vol. 110(1), pages 91-102, September.
    19. Xianyi Wu & Xian Zhou, 2019. "On Hodges’ superefficiency and merits of oracle property in model selection," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1093-1119, October.
    20. Lauren Bin Dong, 2004. "The Behrens-Fisher Problem: An Empirical Likelihood Ratio Approach," Econometrics Working Papers 0404, Department of Economics, University of Victoria.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1095-:d:1370687. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.