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

George Tzavelas

Applications of Mathematics (2012)

  • Volume: 57, Issue: 5, page 531-542
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Tzavelas, George. "A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution." Applications of Mathematics 57.5 (2012): 531-542. <http://eudml.org/doc/246931>.

@article{Tzavelas2012,
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.},
author = {Tzavelas, George},
journal = {Applications of Mathematics},
keywords = {Weibull distribution; Hessian matrix; maximum likelihood estimator; stationary value; Hessian matrix; maximum likelihood estimator; stationary values},
language = {eng},
number = {5},
pages = {531-542},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution},
url = {http://eudml.org/doc/246931},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Tzavelas, George
TI - A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 5
SP - 531
EP - 542
AB - 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.
LA - eng
KW - Weibull distribution; Hessian matrix; maximum likelihood estimator; stationary value; Hessian matrix; maximum likelihood estimator; stationary values
UR - http://eudml.org/doc/246931
ER -

References

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  1. Bain, L. J., Engelhardt, M., Statistical Analysis of Reliability and Life-Testing Models, 2nd ed, Marcel Dekker Inc. New York (1991). (1991) 
  2. Cox, D. R., Oakes, D., Analysis of Survival Data, Chapman Hall London (1984) 0751780. MR0751780
  3. Gourdin, E., Hansen, P., Jaumard, B., 10.1007/BF01096687, J. Glob. Optim. 5 (1994), 373-397. (1994) Zbl0807.62021MR1305084DOI10.1007/BF01096687
  4. Johnson, N. L., Kotz, S., Balakrishnan, N., Continuous Univariate Distributions. Vol. 1, 2nd ed, Wiley Chichester (1994). (1994) Zbl0811.62001MR1299979
  5. Lockhart, R. A., Stephens, M. A., 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 (1993). (1993) MR1278222
  6. Lockhart, R. A., Stephens, M. A., Estimation and tests of fit for the Three-Parameter Weibull Distribution, J. R. Stat. Soc. (Series B) 56 (1994), 491-500. (1994) Zbl0800.62145MR1278222
  7. Marsden, J. E., Tromba, A. J., Vector Calculus, 4th ed, W. H. Freeman New York (1996). (1996) 
  8. McCool, J. I., 10.1109/TR.1970.5216370, IEEE Transactions on Reliability R-19 (1970), 2-9. (1970) DOI10.1109/TR.1970.5216370
  9. Pike, M., 10.2307/2528221, Biometrics 22 (1966), 142-161. (1966) DOI10.2307/2528221
  10. Proschan, F., 10.1080/00401706.1963.10490105, Technometrics 5 (1963), 375-383. (1963) DOI10.1080/00401706.1963.10490105
  11. Qiao, H., Tsokos, C. P., 10.1016/0378-4754(95)95213-5, Math. Comput. Simul. 39 (1995), 173-185 0360857. (1995) MR1360857DOI10.1016/0378-4754(95)95213-5
  12. Rockette, H., Antle, C. E., Klimko, L. A., 10.1080/01621459.1974.10480164, J. Am. Stat. Assoc. 69 (1974), 246-249. (1974) Zbl0283.62033DOI10.1080/01621459.1974.10480164
  13. Smith, R. L., 10.1093/biomet/72.1.67, Biometrika 72 (1985), 67-90. (1985) MR0790201DOI10.1093/biomet/72.1.67
  14. Smith, R. L., Naylor, J. C., 10.1007/BF02054756, Ann. Oper. Res. 9 (1987), 577-587. (1987) DOI10.1007/BF02054756
  15. Smith, R. L., Naylor, J. C., A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, J. R. Stat. Soc., Ser. C 36 (1987), 385-369 0918854. (1987) MR0918854

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