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Testing on the common mean of several normal distributions

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  • 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].

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  • 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
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    References listed on IDEAS

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    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. 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.
    3. 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.
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    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.
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    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.

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