Abstract
The problem of estimating the common mean μ of k independent and univariate inverse Gaussian populations IG(μ, λ i ), i=1,..., k with unknown and unequal λ's is considered. The difficulty with the maximum likelihood estimator of μ is pointed out, and a natural estimator % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] of μ along the lines of Graybill and Deal is proposed. Various finite sample properties and some decision-theoretic properties of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] are discussed.
Similar content being viewed by others
References
Banerjee, A. K. and Bhattacharya, G. K. (1976). A purchase incidence model with inverse Gaussian interpurchase times, J. Amer. Statist. Assoc. 71, 823–829.
Bhattacharya, C. G. (1980). Estimation of a common mean and recovery of interblock information, Ann. Statist., 8, 105–211.
Bravo, G. and MacGibbon, B. (1988). Improved estimation for the parameters of an inverse Gaussian distributin, Comm. Statist. A—Theory Methods, 17, 4285–4299.
Brown, L. D. and Cohen, A. (1974). Point and confidence estimation of a common mean and recovery of inter-block information, Ann. Statist., 2, 963–976.
Chhikara, R. S. and Folks, J. L. (1989). The Inverse Gaussian Distribution, Dekker, New York.
Folks, J. L. and Chhikara, R. S. (1978). The inverse Gaussian distribution and its statistical application, J. Roy. Statist. Soc. Ser. B, 40, 263–289.
Graybill, F. A. and Deal, R. B. (1959). Combining unbiased estimators, Biometrics, 15, 543–550.
Haff, L. R. (1979). An identity for the Wishart distribution with applications, J. Multivariate Anal., 9, 531–544.
Hasofer, A. M. (1964). A dam with inverse Gaussian input, Proc. Camb. Phil. Soc., 60, 931–933.
Hirano, K. and Iwase, K. (1989a). Conditional information for an inverse Gaussian distribution with known coefficient of variation, Ann. Inst. Statist. Math., 41, 279–287.
Hirano, K. and Iwase, K. (1989b). Minimum risk scale equivariant estimator: estimating the mean of an inverse Gaussian distribution with known coefficient of variation, Comm. Statist. A—Theory Methods, 18, 189–197.
Hsieh, H. K., Korwar, R. M. and Rukhin, A. L. (1990). Inadmissibility of the maximum likelihood estimator of the inverse Gaussian mean, Statist. Probab. Lett., 9, 83–90.
Iwase, K. and Seto, N. (1983). Uniformly minimum variance unbiased estimation for the inverse Gaussian distribution, J. Amer. Statist. Assoc., 78, 660–663.
Khatri, C. G. and Shah, K. R. (1974). Estimation of the location parameters from two linear models under normality, Comm. Statist. A—Theory Methods, 3, 647–663.
Korwar, R. M. (1980). On the uniformly minimum variance unbiased estimators of the variance and its reciprocal of an inverse Gaussian distribution, J. Amer. Statist. Assoc., 135, 257–271.
Lancaster, A. (1972). A stochastic model for the duration of a strike, J. Roy. Statist. Soc. Ser. A, 135, 257–271.
Letac, G., Seshadri, V. and Whitmore, G. A. (1985). An exact chi-squared decomposition theorem for inverse Gaussian variates, J. Roy. Statist. Soc. Ser. B, 47, 476–481.
Nair, K. A. (1986). Distribution of an estimator of the common mean of two normal populations, Ann. Statist., 8, 212–216.
Norwood, T. E. and Hinkelmann, K. (1977). Estimating the common mean of several normal populations, Ann. Statist., 5, 1047–1050.
Pal, N. and Sinha, B. K. (1989). Improved estimators of dispersion of an inverse Gaussian distribution, Statistical Data Analysis and Inference, (ed. Y. Dodge), North Holland, Amsterdam.
Pandey, B. N. and Malik, H. J. (1988). Some improved estimators for a measure of dispersion of an inverse Gaussian distribution, Comm. Statist. A—Theory Methods, 17, 3935–3949.
Sheppard, C. W. (1962). Basic Principals of the Tracer Method, Wiley, New York.
Sinha, B. K. (1979). Is the maximum likelihood estimate of the common mean of several normal populations admissible?, Sankhyā Ser. B, 40, 192–196.
Sinha, B. K. and Mouqadem, O. (1982). Estimation of the common mean of two univariate normal populations, Comm. Statist. A—Theory Methods, 11, 1604–1614.
Sinha, B. K. (1985). Unbiased estimation of the variance of the Graybill-Deal estimator of the common mean of several normal populations, Canad. J. Statist., 13, 243–247.
Tweedi, M. C. K. (1957a). Statistical properties of inverse Gaussian distributions I, Ann. Math. Statist., 28, 362–377.
Tweedi, M. C. K. (1957b). Statistical properties of inverse Gaussian distribution II, Ann. Math. Statist., 28, 696–705.
Uspenski, J. V. (1948). Theory of Equations, McGraqhill, New York.
Whitmore, G. A. (1979). An inverse Gaussian model for labour turnover, J. Roy. Statist. Soc. Ser. A, 142, 468–479.
Whitmore, G. A. (1986a). Normal-gamma mixtures of inverse Gaussian distributions, Scand. J. Statist., 13, 211–220.
Whitmore, G. A. (1986b). Inverse Gaussian ratio estimation, Appl. Statist., 35, 8–15.
Zacks, S. (1966). Unbiased estimation of the common mean, J. Amer. Statist. Assoc., 61, 467–476.
Zacks, S. (1970). Bayes and fiducial equivariant estimators of the common mean of two normal distributions, Ann. Math. Statist., 41, 59–69.
Author information
Authors and Affiliations
Additional information
This research was partially supported by research grants #A3661 and #A3450 from NSERC of Canada.
About this article
Cite this article
Ahmad, M., Chaubey, Y.P. & Sinha, B.K. Estimation of a common mean of several univariate inverse Gaussian populations. Ann Inst Stat Math 43, 357–367 (1991). https://doi.org/10.1007/BF00118641
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00118641