Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
권호 표지

SOME POINT ESTIMATES FOR THE SHAPE PARAMETERS OF EXPONENTIATED-WEIBULL FAMILY

  • Singh Umesh (Department of Statistics, Banaras Hindu University) ;
  • Gupta Pramod K. (Department of Statistics, Banaras Hindu University, Division of Biostatistics and Bioinformatics, National Health Research Institutes) ;
  • Upadhyay S.K. (Department of Statistics, Banaras Hindu University)
  • Published : 2006.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Maximum product of spacings estimator is proposed in this paper as a competent alternative of maximum likelihood estimator for the parameters of exponentiated-Weibull distribution, which does work even when the maximum likelihood estimator does not exist. In addition, a Bayes type estimator known as generalized maximum likelihood estimator is also obtained for both of the shape parameters of the aforesaid distribution. Though, the closed form solutions for these proposed estimators do not exist yet these can be obtained by simple appropriate numerical techniques. The relative performances of estimators are compared on the basis of their relative risk efficiencies obtained under symmetric and asymmetric losses. An example based on simulated data is considered for illustration.

Keywords

References

  1. BAIN, L. J. (1974). 'Analysis for the linear failure-rate life-testing distribution', Technometrics, 16, 551-559 https://doi.org/10.2307/1267607
  2. CHENG, R. C. H. AND AMIN, N. A. K. (1983). 'Estimating parameters in continuous univariate distributions with a shifted origin', Journal of the Royal Statistical Society, Ser. B, 45, 394-403
  3. GORE, A. P., PARANJAPE, S. A., RAJARSHI, M. B. AND GADGIL, M. (1986). 'Some methods for summarising survivorship data in nonstandard situations', Biometrical Journal, 28, 557-586
  4. HARTER, H. L. AND MOORE, A. H. (1966). 'Local-maximum-likelihood estimation of the parameters of the three-parameter lognormal populations from complete and censored samples', Journal of the American Statistical Association, 61, 842-851 https://doi.org/10.2307/2282793
  5. HOSSAIN, A. M. AND NATH, S. K. (1997). 'Estimation of parameters in the presence of outliers for a Burr XII distribution', Communications in Statistics- Theory and Methods, 26, 637-652 https://doi.org/10.1080/03610929708831939
  6. HUZURBAZAR, V. S. (1948). 'The likelihood equation, consistency and the maxima of the likelihood function', Annals of Eugenics, 14, 185-200
  7. JOHNSON, N. L. AND KOTZ, S. (1970). Distributions in Statistics. Continuous Univariate Distributions 1., Houghton Mifflin, Boston
  8. KHATREE, R. (1992). 'Estimation of gurantee time and mean after warranty for two-parameter exponential failure model', The Australian Journal of Statistics, 34, 207-215 https://doi.org/10.1111/j.1467-842X.1992.tb01354.x
  9. LAWLESS, J. F. (1982). Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York
  10. MARTZ, H. F. AND WALLER, R. A. (1982). Bayesian Reliability Analysis, John Wiley & Sons, Chichester
  11. MUDHOLKAR, G. S. AND KOLLIA, G. D. (1990). 'Isotones of the test of exponentially', Proceedings of the Statistical Graphics Section, 82-87
  12. MUDHOLKAR, G. S., KOLLIA, G. D., LIN, C. T. AND PATEL, K. R. (1991). 'A graphical procedure for comparing goodness-of-fit tests', Journal of the Royal Statistical Society, Ser. B, 53, 221-232
  13. MUDHOLKAR, G. S. AND HUTSON A. D. (1996). 'The exponentiated Weibull family: Some properties and flood data application', Communications in Statistics-Theory and Methods, 25, 3059-3083 https://doi.org/10.1080/03610929608831886
  14. MUDHOLKAR, G. S., SRIVASTAVA, D. K. AND KOLLIA, G. D. (1996). 'A generalization of Weibull distribution with application to the analysis of survival data', Journal of the American Statistical Association, 91, 1575-1583 https://doi.org/10.2307/2291583
  15. NAG (1993). Mark 16, Numerical Algorithms Group, Illinois
  16. PRENTICE, R. L. (1975). 'Discrimination among some parametric models', Biometrika, 62, 607-614 https://doi.org/10.1093/biomet/62.3.607
  17. RAJARSHI, S. AND RAJARSHI, M. B. (1988). 'Bathtub distributions: A review', Communications in Statistics-Theory and Methods, 17, 2597-2621 https://doi.org/10.1080/03610928808829761
  18. RANNEBY, BO (1984). 'The maximum spacings method. An estimation method related to maximum likelihood method', Scandinavian Journal of Statistics-Theory and Applications, 11, 93-112
  19. ROJO, J. (1987). 'On the admissibility of c$\overline{X}$ + d with respect to the LINEX loss function', Communications in Statistics-Theory and Methods, 16, 3745-3748 https://doi.org/10.1080/03610928708829603
  20. SHAH, A. AND GOKHALE, D. V. (1993). 'On maximum product of spacings (MPS) estimation for Burr XII distributions', Communications in Statistics-Simulation and Computation, 22, 615-644 https://doi.org/10.1080/03610919308813112
  21. SINHA, S. K. (1986) Reliability and Life Testing. With a Foreword by C. R. Rao., John Wiley & Sons, New York
  22. SINGH, U., GUPTA, PRAMOD K., CHATURVADI, M. AND UPADHYAY, S. K. (1999). 'Bayesian estimation of exponentiated- Wei bull distribution using squared error loss function', Proceeding of National Seminar on Bayesian Analysis-Theory and Application, 130-144
  23. SINGH, U., GUPTA, PRAMOD K. AND UPADHYAY, S. K. (2002). 'Estimation of exponentiated Weibull shape parameters under LINEX loss function', Communication of Statistics-Simulation and Computation, 31, 523-537 https://doi.org/10.1081/SAC-120004310
  24. SINGH, U., GUPTA, PRAMOD K. AND UPADHYAY, S. K. (2005a). 'Estimation of parameters for exponentiated- Weibull family under type-II censoring scheme', Computational Statistics and Data Analysis, 48, 509-523 https://doi.org/10.1016/j.csda.2004.02.009
  25. SINGH, U., GUPTA, PRAMOD K. AND UPADHYAY, S. K. (2005b). 'Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring', Journal of Statistical Planning and Inference, 134, 350-372 https://doi.org/10.1016/j.jspi.2004.04.018
  26. STACY, E. W. (1962). 'A generalization of the gamma distribution', Annals of Mathematical Statistics, 33, 1187-1192 https://doi.org/10.1214/aoms/1177704481
  27. SLYMEN, D. J. AND LACHENBRUCH, P. A. (1984). 'Survival distributions arising from two families and generated by transformation', Communications in Statistics- Theory and Methods, 13, 1179-1201 https://doi.org/10.1080/03610928408828748
  28. VARIAN, H. R. (1975). 'A Bayesian approach to real estate assessment', In Studies in Bayesian Econometrics and Statistics: in Honor of Leonard J. Savage (S. E. Fienberg and A. Zellner, eds.), 195-208, North-Holland, Amsterdam
  29. ZELLNER, ARNOLD (1986). 'Bayesian estimation and prediction using asymmetric loss functions', Journal of the American Statistical Association, 81, 446-451 https://doi.org/10.2307/2289234