Previous |  Up |  Next


Title: A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution (English)
Author: Tzavelas, George
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 57
Issue: 5
Year: 2012
Pages: 531-542
Summary lang: English
Category: math
Summary: 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. (English)
Keyword: Weibull distribution
Keyword: Hessian matrix
Keyword: maximum likelihood estimator
Keyword: stationary value
MSC: 62F10
MSC: 62F99
MSC: 62N05
idZBL: Zbl 1263.62034
idMR: MR2984618
DOI: 10.1007/s10492-012-0031-x
Date available: 2012-08-19T22:09:09Z
Last updated: 2020-07-02
Stable URL:
Reference: [1] Bain, L. J., Engelhardt, M.: Statistical Analysis of Reliability and Life-Testing Models, 2nd ed.Marcel Dekker Inc. New York (1991).
Reference: [2] Cox, D. R., Oakes, D.: Analysis of Survival Data.Chapman & Hall London (1984) \MR 0751780. MR 0751780
Reference: [3] Gourdin, E., Hansen, P., Jaumard, B.: Finding maximum likelihood estimators for the three-parameter Weibull distribution.J. Glob. Optim. 5 (1994), 373-397. Zbl 0807.62021, MR 1305084, 10.1007/BF01096687
Reference: [4] Johnson, N. L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions. Vol. 1, 2nd ed.Wiley Chichester (1994). Zbl 0811.62001, MR 1299979
Reference: [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). MR 1278222
Reference: [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. Zbl 0800.62145, MR 1278222
Reference: [7] Marsden, J. E., Tromba, A. J.: Vector Calculus, 4th ed.W. H. Freeman New York (1996).
Reference: [8] McCool, J. I.: Inference on Weibull percentiles and shape parameter from maximum likelihood estimates.IEEE Transactions on Reliability R-19 (1970), 2-9. 10.1109/TR.1970.5216370
Reference: [9] Pike, M.: A suggested method of analysis of a certain class of experiments in carcinogenesis.Biometrics 22 (1966), 142-161. 10.2307/2528221
Reference: [10] Proschan, F.: Theoretical explanation of observed decreasing failure rate.Technometrics 5 (1963), 375-383. 10.1080/00401706.1963.10490105
Reference: [11] Qiao, H., Tsokos, C. P.: Estimation of the three parameter Weibull probability distribution.Math. Comput. Simul. 39 (1995), 173-185 \MR 0360857. MR 1360857, 10.1016/0378-4754(95)95213-5
Reference: [12] Rockette, H., Antle, C. E., Klimko, L. A.: Maximum likelihood estimation with the Weibull model.J. Am. Stat. Assoc. 69 (1974), 246-249. Zbl 0283.62033, 10.1080/01621459.1974.10480164
Reference: [13] Smith, R. L.: Maximum likelihood estimation in a class of non-regular cases.Biometrika 72 (1985), 67-90. MR 0790201, 10.1093/biomet/72.1.67
Reference: [14] Smith, R. L., Naylor, J. C.: Statistics of the three-parameter Weibull distribution.Ann. Oper. Res. 9 (1987), 577-587. 10.1007/BF02054756
Reference: [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 \MR 0918854. MR 0918854


Files Size Format View
AplMat_57-2012-5_7.pdf 251.7Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo