Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Bayesian Parameter Estimation of the Four-Parameter Gamma Distribution

  • Oh, Mi-Ra (Department of Statistics, Chonnam National University) ;
  • Kim, Kyung-Sook (Department of Statistics, Chonnam National University) ;
  • Cho, Wan-Hyun (Department of Statistics, Chonnam National University) ;
  • Son, Young-Sook (Department of Statistics, Chonnam National University)
  • Published : 2007.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

A Bayesian estimation of the four-parameter gamma distribution is considered under the noninformative prior. The Bayesian estimators are obtained by the Gibbs sampling. The generation of the shape/power parameter and the power parameter in the Gibbs sampler is implemented using the adaptive rejection sampling algorithm of Gilks and Wild (1992). Also, the location parameter is generated using the adaptive rejection Metropolis sampling algorithm of Gilks, Best and Tan (1995). Finally, the simulation result is presented.

Keywords

References

  1. Abramowitz, M. and Stegun, I. A. (1972). Polygamma Functions. Handbook of Mathematical Functions with Formulas, Graph, and Mathematical Tables, 9th printing, Dover, New York
  2. Bowman, K. O. and Shenton, L. R. (1988). Properties of Estimators for the Gamma Distribution. Marcel Dekker, New York
  3. Cohen, A .C. and Norgaard, N. J. (1977). Progressively censored sampling in the three-parameter gamma distribution. Technometrics, 19,333-340 https://doi.org/10.2307/1267704
  4. Cohen, A. C. and Whitten, B. J. (1982). Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution. Communications in Statistics-Simulation and Computation. 11, 197-216 https://doi.org/10.1080/03610918208812254
  5. Cohen, A. C. and Whitten, B. J. (1986). Modified moment estimation for the three-parameter gamma distribution. Journal of Quality Technology, 18, 53-62 https://doi.org/10.1080/00224065.1986.11978985
  6. Gilks, W. R., Best, N. G. and Tan, K. K. C. (1995). Adaptive rejection Metropolis sampling within Gibbs sampling. Applied Statistics, 44, 455-472 https://doi.org/10.2307/2986138
  7. Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337-348 https://doi.org/10.2307/2347565
  8. Greenwood, J. A. and Durand, D. (1960). Aids for fitting the gamma distribution by maximum likelihood. Technometrics, 2, 55-65 https://doi.org/10.2307/1266531
  9. Hager, H. W. and Bain, L. J. (1970). Inferential procedures for the generalized gamma distribution. Journal of the American Statistical Association, 65, 1601-1609 https://doi.org/10.2307/2284343
  10. Hager, H. W., Bain, L. J. and Antle, C. E. (1971). Reliability estimation for the generalized gamma distribution and robustness of the Weibull model. Technometrics, 13, 547-557 https://doi.org/10.2307/1267167
  11. Harter, H. L. (1967). Maximum-likelihood estimation of the parameters of a four-parameter generalized gamma population from complete and censored samples. Technometrics, 9, 159-165 https://doi.org/10.1080/00401706.1967.10490449
  12. Harter, H. L. and Moore, A. H. (1965). Maximum-likelihood estimation of the parameters of gamma and Weibull populations from complete and from censored samples. Technometrics, 7, 639-643 https://doi.org/10.2307/1266401
  13. Johnson, N. L. and Kotz, S. (1970). Distributions in Statistics: Continuous Univariate Distributions-1. John Wiley & Sons, New York
  14. Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics, 22, 409-419 https://doi.org/10.2307/1268326
  15. Parr, V. B. and Webster, J. T. (1965). A method for the discriminating between failure density functions used in reliability predictions. Technometrics, 7, 1-10 https://doi.org/10.2307/1266123
  16. Pang, W. K., Hou, S. H., Yu, B. W. T. and Li, K. W. K. (2004). A simulation based approach to the parameter estimation for the three-parameter gamma distribution. European Journal of Operational Research, 155, 675-682 https://doi.org/10.1016/S0377-2217(02)00883-4
  17. Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika, 61, 539-544 https://doi.org/10.1093/biomet/61.3.539
  18. Ripley, B. D. (1988). Stochastic Simulation. John Wiley & Sons, New York
  19. Ritter, C. and Tanner, M. A. (1992). Facilitating the Gibbs sampler: the Gibbs stopper and the griddy-Gibbs sampler. Journal of the American Statistical Association, 87, 861-868 https://doi.org/10.2307/2290225
  20. Roberts, G. O. and Smith, A. F. M. (1994). Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms. Stochastic Processes and their Applications, 49, 207-216 https://doi.org/10.1016/0304-4149(94)90134-1
  21. Son, Y, Sand Oh, M. (2006). Bayesian estimation of the two-parameter gamma distribution. Communications in Statistics-Simulation and Computation, 35, 285-293 https://doi.org/10.1080/03610910600591925
  22. Stacy, E. W. (1973). Quasimaximum likelihood estimators for two-parameter gamma distributions. IBM Journal of Research and Development, 17, 115-124 https://doi.org/10.1147/rd.172.0115
  23. Stacy, E. W. and Mihram, G. A. (1965). Parameter estimation for a generalized gamma distribution. Technometrics, 7, 349-358 https://doi.org/10.2307/1266594
  24. The MathWorks Inc. (2002). MATLAB/Statistics Toolbox, Version 6.5. Natick, MA
  25. Thom, H. C. S. (1958). A note on the gamma distribution. Monthly Weather Review, 86,117-122 https://doi.org/10.1175/1520-0493(1958)086<0117:ANOTGD>2.0.CO;2
  26. Tsionas, E. G. (2001). Exact inference in four-parameter generalized gamma distribution. Communications in Statistics-Theory and Methods, 30, 747-756 https://doi.org/10.1081/STA-100002149