Skip to main content
SpringerLink
Log in

A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

The maximum likelihood estimators of the parameters for the 3-parameter Weibull distribution do not always exist. Furthermore, computationally it is difficult to find all the solutions. Thus, the case of missing some solutions and among them the maximum likelihood estimators cannot be excluded. In this paper we provide a simple rule with help of which we are able to know if the system of the log-likelihood equations has even or odd number of solutions. It is a useful tool for the detection of all the solutions of the system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. J. Bain, M. Engelhardt: Statistical Analysis of Reliability and Life-Testing Models, 2nd ed. Marcel Dekker Inc., New York, 1991.

    MATH  Google Scholar 

  2. D.R. Cox, D. Oakes: Analysis of Survival Data. Chapman & Hall, London, 1984.

    Google Scholar 

  3. E. Gourdin, P. Hansen, B. Jaumard: Finding maximum likelihood estimators for the three-parameter Weibull distribution. J. Glob. Optim. 5 (1994), 373–397.

    Article  MathSciNet  MATH  Google Scholar 

  4. N. L. Johnson, S. Kotz, N. Balakrishnan: Continuous Univariate Distributions. Vol. 1, 2nd ed. Wiley, Chichester, 1994.

    MATH  Google Scholar 

  5. R.A. Lockhart, M.A. Stephens: Estimation and Tests of Fit for the Three-Parameter Weibull Distribution. Research Report 92-10 (1993). Department of Mathematics and Statistics, Simon Frasher University, Burnaby.

  6. R.A. Lockhart, M.A. Stephens: Estimation and tests of fit for the Three-Parameter Weibull Distribution. J. R. Stat. Soc. (Series B) 56 (1994), 491–500.

    MathSciNet  MATH  Google Scholar 

  7. J.E. Marsden, A. J. Tromba: Vector Calculus, 4th ed. W.H. Freeman, New York, 1996.

    MATH  Google Scholar 

  8. J. I. McCool: Inference on Weibull percentiles and shape parameter from maximum likelihood estimates. IEEE Transactions on Reliability R-19 (1970), 2–9.

    Article  Google Scholar 

  9. M. Pike: A suggested method of analysis of a certain class of experiments in carcinogenesis. Biometrics 22 (1966), 142–161.

    Article  Google Scholar 

  10. F. Proschan: Theoretical explanation of observed decreasing failure rate. Technometrics 5 (1963), 375–383.

    Article  Google Scholar 

  11. H. Qiao, C.P. Tsokos: Estimation of the three parameter Weibull probability distribution. Math. Comput. Simul. 39 (1995), 173–185.

    Article  MathSciNet  Google Scholar 

  12. H. Rockette, C.E. Antle, L.A. Klimko: Maximum likelihood estimation with theWeibull model. J. Am. Stat. Assoc. 69 (1974), 246–249.

    Article  MATH  Google Scholar 

  13. R. L. Smith: Maximum likelihood estimation in a class of non-regular cases. Biometrika 72 (1985), 67–90.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. L. Smith, J.C. Naylor: Statistics of the three-parameter Weibull distribution. Ann. Oper. Res. 9 (1987), 577–587.

    Article  Google Scholar 

  15. R. L. Smith, J.C. Naylor: A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. J. R. Stat. Soc., Ser. C 36 (1987), 385–369.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou Street, 185 34, Piraeus, Greece

    George Tzavelas

Authors
  1. George Tzavelas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Tzavelas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tzavelas, G. A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution. Appl Math 57, 531–542 (2012). https://doi.org/10.1007/s10492-012-0031-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10492-012-0031-x

Keywords

MSC 2010

Search

Navigation