IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v53y2008i2p321-333.html
   My bibliography  Save this article

Testing on the common mean of several normal distributions

Author

Listed:
  • Chang, Ching-Hui
  • Pal, Nabendu

Abstract

Point estimation of the common mean of several normal distributions with unknown and possibly unequal variances has attracted the attention of many researchers over the last five decades. Relatively less attention has been paid to the hypothesis testing problem, presumably due to the complicated sampling distribution(s) of the test statistics(s) involved. Taking advantage of the computational resources available nowadays there has been a renewed interest in this problem, and a few test procedures have been proposed lately including those based on the generalized p-value approach. In this paper we propose three new tests based on the famous Graybill-Deal estimator (GDE) as well as the maximum likelihood estimator (MLE) of the common mean, and these test procedures appear to work as good as (if not better than) the existing test methods. The two tests based on the GDE use respectively a first order unbiased variance estimate proposed by Sinha [Sinha, B.K., 1985. Unbiased estimation of the variance of the Graybill-Deal estimator of the common mean of several normal populations. The Canadian Journal of Statistics 13 (3), 243-247], as well as the little known exact unbiased variance estimator proposed by Nikulin and Voinov [Nikulin, M.S., Voinov, V.G., 1995. On the problem of the means of weighted normal populations. Qüestiió (Quaderns d'Estadistica, Sistemes, Informatica i Investigació Operativa) 19 (1-3), 93-106] (after we've fixed a small mistake in the final expression). On the other hand, the MLE, which doesn't have a closed expression, uses a parametric bootstrap method proposed by Pal, Lim and Ling [Pal, N., Lim, W.K., Ling, C.H., 2007b. A computational approach to statistical inferences. Journal of Applied Probability & Statistics 2 (1), 13-35]. The extensive simulation results presented in this paper complement the recent studies undertaken by Krishnamoorthy and Lu [Krishnamoorthy, K., Lu, Y., 2003. Inferences on the common mean of several normal populations based on the generalized variable method. Biometrics 59, 237-247], and Lin and Lee [Lin, S.H., Lee, J.C., 2005. Generalized inferences on the common mean of several normal populations. Journal of Statistical Planning and Inference 134, 568-582].

Suggested Citation

  • 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.
    • Handle: RePEc:eee:csdana:v:53:y:2008:i:2:p:321-333
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-9473(08)00353-8
    Download Restriction: Full text for ScienceDirect subscribers only.
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Haff, L. R., 1979. "An identity for the Wishart distribution with applications," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 531-544, December.
    2. K. Krishnamoorthy & Yong Lu, 2003. "Inferences on the Common Mean of Several Normal Populations Based on the Generalized Variable Method," Biometrics, The International Biometric Society, vol. 59(2), pages 237-247, June.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Elena Kulinskaya & Stephan Morgenthaler & Robert G. Staudte, 2014. "Combining Statistical Evidence," International Statistical Review, International Statistical Institute, vol. 82(2), pages 214-242, August.
    3. Liu, Weihua & Liu, Xiaoyan & Li, Xiang, 2015. "The two-stage batch ordering strategy of logistics service capacity with demand update," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 83(C), pages 65-89.
    4. Malekzadeh, Ahad & Kharrati-Kopaei, Mahmood, 2017. "An exact method for making inferences on the common location parameter of several heterogeneous exponential populations: Complete and censored data," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 210-215.
    5. Ahad Malekzadeh & Mahmood Kharrati-Kopaei, 2018. "Inferences on the common mean of several normal populations under heteroscedasticity," Computational Statistics, Springer, vol. 33(3), pages 1367-1384, September.
    6. Sang Gil Kang & Woo Dong Lee & Yongku Kim, 2017. "Objective Bayesian testing on the common mean of several normal distributions under divergence-based priors," Computational Statistics, Springer, vol. 32(1), pages 71-91, March.

    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. 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.
    2. Andrew F. Siegel & Artemiza Woodgate, 2007. "Performance of Portfolios Optimized with Estimation Error," Management Science, INFORMS, vol. 53(6), pages 1005-1015, June.
    3. Kubokawa, Tatsuya & Srivastava, Muni S., 2008. "Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1906-1928, October.
    4. K. Krishnamoorthy, 1991. "Estimation of a common multivariate normal mean vector," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(4), pages 761-771, December.
    5. Francesco Lautizi, 2015. "Large Scale Covariance Estimates for Portfolio Selection," CEIS Research Paper 353, Tor Vergata University, CEIS, revised 07 Aug 2015.
    6. Konno, Yoshihiko, 2009. "Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2237-2253, November.
    7. Tsukada, Shin-ichi, 2014. "Asymptotic expansion for distribution of the trace of a covariance matrix under a two-step monotone incomplete sample," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 206-219.
    8. Elfessi, Abdulaziz & Chun Jin, 1996. "On robust estimation of the common scale parameter of several Pareto distributions," Statistics & Probability Letters, Elsevier, vol. 29(4), pages 345-352, September.
    9. Li, Xinmin & Wang, Juan & Liang, Hua, 2011. "Comparison of several means: A fiducial based approach," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1993-2002, May.
    10. Wang, Dan & Tian, Lili, 2017. "Parametric methods for confidence interval estimation of overlap coefficients," Computational Statistics & Data Analysis, Elsevier, vol. 106(C), pages 12-26.
    11. 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.
    12. Tsai, Ming-Tien, 2007. "Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 932-944, May.
    13. H. Zakerzadeh & A. Jafari, 2015. "Inference on the parameters of two Weibull distributions based on record values," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(1), pages 25-40, March.
    14. Tsukuma, Hisayuki, 2016. "Estimation of a high-dimensional covariance matrix with the Stein loss," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 1-17.
    15. Kubokawa, Tatsuya & Tsai, Ming-Tien, 2006. "Estimation of covariance matrices in fixed and mixed effects linear models," Journal of Multivariate Analysis, Elsevier, vol. 97(10), pages 2242-2261, November.
    16. Tsukuma, Hisayuki, 2010. "Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1483-1492, July.
    17. Tsai, Ming-Tien & Kubokawa, Tatsuya, 2007. "Estimation of Wishart mean matrices under simple tree ordering," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 945-959, May.
    18. Bodnar, Olha & Bodnar, Taras & Parolya, Nestor, 2022. "Recent advances in shrinkage-based high-dimensional inference," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    19. Shokofeh Zinodiny & Saralees Nadarajah, 2024. "A New Class of Bayes Minimax Estimators of the Mean Matrix of a Matrix Variate Normal Distribution," Mathematics, MDPI, vol. 12(7), pages 1-14, April.
    20. Tsukuma, Hisayuki & Konno, Yoshihiko, 2006. "On improved estimation of normal precision matrix and discriminant coefficients," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1477-1500, August.

    More about this item

    Statistics

    Access and download statistics

    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:eee:csdana:v:53:y:2008:i:2:p:321-333. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    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.