Testing on the common mean of several normal distributions

Ching-Hui Chang and Nabendu Pal

Computational Statistics & Data Analysis, 2008, vol. 53, issue 2, 321-333

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].

Date: 2008
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