A Note on the Estimation of Restricted Scale Parameters of Gamma Distributions

  • Yuan-Tsung Chang


In this paper an admissible estimator of the scale parameter of a Gamma distribution is derived, which is bounded above. Next, the simultaneous estimation of p scale parameters of Gamma distributions are considered that are bounded above. Moreover, simultaneous estimation of two ordered scale parameters of two Gamma distributions is investigated in terms of the mean square error (MSE). Finally, some simulation results are given for illustrating the improvement obtained with the new estimators, when compared with the usual estimators.


Scale Parameter Gamma Distribution Maximum Likelihood Estimator Simultaneous Estimation Order Restriction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author would like to thank Prof. Nobuo Shinozaki, Keio University for many discussions, suggestions and for checking the manuscript.


  1. 1.
    Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions, Wiley, New York.MATHGoogle Scholar
  2. 2.
    Berger, J. (1980). Improving on inadmissible estimators in continuous exponential families with applications to simultaneous estimation of gamma scale parameters. Ann. Statist., 8, 545-571.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berry, J.C. (1993). Minimax estimation of a restricted exponential location parameter. Statist. Decisions, 11, 307-316.MATHMathSciNetGoogle Scholar
  4. 4.
    Casella, G. and Strawderman, W.E (1981). Estimating a bounded normal mean. Ann. Statist., 9, 870-878.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chang, Y.-T. (1982). Stein-type estimators for parameters in truncated spaces. Keio Science and Technology Reports, 34, 83-95.Google Scholar
  6. 6.
    Chang, Y.-T. and Shinozaki, N. (2002). A comparison of restricted and unrestricted estimators in estimating linear functions of ordered scale parameters of two gamma distributions. Ann. Inst. Statist. Math. Vol. 54, No.4, 848-860.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Katz,M.W. (1961). Admissible and minimax estimates of parameters in truncated spaces. Ann. Math. Statist., 32, 136-142.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Marchand, È. and Strawderman, W.E. (2005) On improving on the minimum risk equivariant estimator of a scale parameter under a lower-bound constraint. Journal of Statist. planning and inference, 134. 90-101.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Robertson, T., Wright, F. T. and Dykstra, R. L.(1988). Order Restricted Statistical Inference, Wiley, New York.MATHGoogle Scholar
  10. 10.
    Shinozaki, N. and Chang, Y.-T. (1999). A comparison of maximum likelihood and best unbiased estimators in the estimation of linear combinations of positive normal means. Statistics & Decisions, 17, 125-136.MATHMathSciNetGoogle Scholar
  11. 11.
    Silvapulle, M. J. and Sen, P. K. (2004). Constrained Statistical Inference, Wiley, New Jersey.Google Scholar
  12. 12.
    Van Eeden,C. (1995). Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss. Canad.J.Statist., 23, 245-256.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Van Eeden, C (2006). Restricted parameter space estimation problems. Lecture notes in Statistics 188, Springer.Google Scholar

Copyright information

© Physica-Verlag Heidelberg 2010

Authors and Affiliations

  1. 1.Dept. of Social Information, Faculty of Studies on Contemporary SocietyMejiro UniversityTokyoJapan

Personalised recommendations