Abstract
In this paper, we consider the problem of making inferences on the common mean of several normal populations when sample sizes and population variances are possibly unequal. We are mainly concerned with testing hypothesis and constructing confidence interval for the common normal mean. Several researchers have considered this problem and many methods have been proposed based on the asymptotic or approximation results, generalized inferences, and exact pivotal methods. In addition, Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) proposed a parametric bootstrap (PB) approach for this problem based on the maximum likelihood estimators. We also propose a PB approach for making inferences on the common normal mean under heteroscedasticity. The advantages of our method are: (i) it is much simpler than the PB test proposed by Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) since our test statistic is not based on the maximum likelihood estimators which do not have explicit forms, (ii) inverting the acceptance region of test yields a genuine confidence interval in contrast to some exact methods such as the Fisher’s method, (iii) it works well in terms of controlling the Type I error rate for small sample sizes and the large number of populations in contrast to Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) method, (iv) finally, it has higher power than recommended methods such as the Fisher’s exact method.
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References
Brillinger DR (1962) A note on the rate of convergence of a mean. Biometrika 49:574–576
Chang CH, Pal N (2008) Testing on the common mean of several normal distributions. Comput Stat Data Anal 53:321–333
Cohen A, Sackrowitz HB (1984) Testing hypotheses about the common mean of normal distributions. J Statist Plan Inference 9:207–227
Eberhardt KR, Reeve CP, Spiegelman CH (1989) A minimax approach to combining means, with practical examples. Chemom Intell Lab Syst 5:129–148
Fairweather WR (1972) A method of obtaining an exact confidence interval for the common mean of several normal populations. Appl Stat 21:229–233
Fisher RA (1932) Statistical methods for research workers, 4th edn. Oliver and Boyd, London
Graybill FA, Deal RB (1959) Combining unbiased estimators. Biometrics 15:543–550
Hannig J, Iyer H, Patterson P (2006) Fiducial generalized confidence intervals. J Am Stat Assoc 101:254–269
Hartung J, Knapp G (2005) Models for combining results of different experiments: retrospective and prospective. Am J Math Manag Sci 25:149–188
Hartung J, Knapp G (2009) Exact and generalized confidence intervals in the common mean problem. In: Schipp B, Kräer W (eds) Statistical Inference, Econometric Analysis and Matrix Algebra, Physica-Verlag HD, pp 85–102
Hartung J, Knapp G, Sinha BK (2008) Statistical meta-analysis with applications. Wiley, New York
Hedges LV, Olkin I (1985) Statistical methods for meta-analysis. Academic, Boston
Jordan SM, Krishnamoorthy K (1996) Exact confidence intervals for the common mean of several normal populations. Biometrics 52:77–86
Krishnamoorthy K, Lu Y (2003) Inferences on the common mean of several normal populations based on the generalized variable method. Biometrics 59:237–247
Lin SH, Lee JC (2005) Generalized inferences on the common mean of several normal populations. J Stat Plan Inference 134:568–582
Maric N, Graybill FA (1979) Small samples confidence intervals on common mean of two normal distributions with unequal variances. Commun Stat Theory Methods 8:1255–1269
Meier P (1953) Variance of a weighted mean. Biometrics 9:59–73
Mitra PK, Sinha BK (2007) On some aspects of estimation of a common mean of two independent normal populations. J Stat Plan Inference 137:184–193
Nasri A, Seyed Hosseini SJ (2014) The comparison of hepatic and biliary duct sonography accuracy performed by the emergency medicine and radiology residents in patients with right upper quadrant abdominal pain. Board Thesis, Tehran University of medical sciences
Nikulin MS, Voinov VG (1995) On the problem of the means of weighted normal populations. Qüestiió 19:93–106
Pagurova VI, Gurskii VV (1979) A confidence interval for the common mean of several normal distributions. Theoy Prob Appl 24:882–888
Pal N, Sinha BK (1996) Estimation of a common mean of several normal populations: a review. Far East J Math Sci I:97–110
Rukhin A (2017) Estimation of the common mean from heterogeneous normal observations with unknown variances. J R Stat Soc B 5:1601–1618
Sinha BK (1985) Unbiased estimation of the variance of the Graybill–Deal estimator of the common mean of several normal populations. Can J Stat 13:243–247
Suguira N, Gupta AK (1987) Maximum likelihood estimates for Behrens–Fisher problem. J Jpn Stat Soc 17:55–60
Tippett LH (1931) The method of statistics. Williams and Norgate, London
Yu PLH, Sun Y, Sinha BK (1999) On exact confidence intervals for the common mean of several normal populations. J Stat Plan Inference 81:263–277
Weerahandi S (1993) Generalized confidence intervals. J Am Stat Assoc 88:899–905
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The authors would like to thank the referees for their constructive comments. The second author would like to acknowledge the Research Council of Shiraz University.
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Malekzadeh, A., Kharrati-Kopaei, M. Inferences on the common mean of several normal populations under heteroscedasticity. Comput Stat 33, 1367–1384 (2018). https://doi.org/10.1007/s00180-017-0789-0
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DOI: https://doi.org/10.1007/s00180-017-0789-0