Finding maximum likelihood estimators for the three-parameter Weibull distribution. (English) Zbl 0807.62021

Summary: Much work has been devoted to the problem of finding maximum likelihood estimators for the three-parameter Weibull distribution. This problem has not been clearly recognized as a global optimization one and most methods from the literature occasionally fail to find a global optimum. We develop a global optimization algorithm which uses first order conditions and projection to reduce the problem to a univariate optimization one. Bounds on the resulting function and its first order derivative are obtained and used in a branch-and-bound scheme. Computational experience is reported. It is also shown that the solution method we propose can be extended to the case of right censored samples.


62F10 Point estimation
90C90 Applications of mathematical programming
90C30 Nonlinear programming
65C99 Probabilistic methods, stochastic differential equations
Full Text: DOI


[1] Adatia, A. and Chan, L.K. (1985), Robust Estimators of the 3-Parameter Weibull Distribution,IEEE Transactions on Reliability 34(4), 347-351. · Zbl 0587.62074
[2] Arnaudies, J.M. and Fraysse, H. (1989),Cours de Mathématiques ? 3: Compléments d’analyse, Dunod Université, Paris.
[3] Atkinson, M.D., Sack, J.-R., Santoro, N. and Strothotte, T. (1986), Min-Max Heaps and Generalized Priority Queues,Communications of the ACM 29(10), 996-1000. · Zbl 0642.68055
[4] Bain, L.J. (1978), Statistical Analysis of Reliability and Life-Testing Models: Theory and Methods,Statistics: Textbooks and Monographs 24, Marcel Dekker Inc., New York. · Zbl 0419.62076
[5] Cohen, A.C., Whitten, B.J. and Ding, Y. (1984), Modified Moment Estimation for the Three-Parameter Weibull Distribution,Journal of Quality Technology 16(3), 159-167.
[6] Dumonceaux, R. and Antle, C.E. (1973), Discrimination between the Log-Normal and the Weibull Distribution,Technometrics 15, 923-926. · Zbl 0269.62024
[7] Geoffrion, A.M. (1972), Generalized Benders Decomposition,Journal of Optimization Theory and Applications 10(4), 237-260. · Zbl 0229.90024
[8] Hansen, P., Jaumard, B. and Lu, S.-H. (1992), Global Optimization of Univariate Lipschitz Functions: I. Survey and Properties,Mathematical Programming 55, 251-272. · Zbl 0825.90755
[9] Hansen, P., Jaumard, B. and Lu, S.-H. (1992), Global Optimization of Univariate Lipschitz Functions: II. New Algorithms and Computational Comparison,Mathematical Programming 55, 273-292. · Zbl 0825.90756
[10] Harter, H.L. (1971), Some Optimization Problems in Parameter Estimation,Optimizing Methods in Statistics, 33-62. · Zbl 0263.62021
[11] Hasham, A. and Sack, J.-R. (1987), Bounds for Min-Max Heaps,BIT 27, 315-323. · Zbl 0642.68056
[12] Horst, R. and Tuy, H. (1987), On the Convergence of Global Methods in Multiextremal Optimization,Journal of Optimization Theory and Applications 54, 253-271. · Zbl 0595.90079
[13] Horst, R. and Tuy, H. (1990),Global Optimization -Deterministic Approaches, Springer-Verlag, New York (2nd edition 1992). · Zbl 0704.90057
[14] Kotz, S. and Johnson, N.L. (1973), Statistical Distributions: Survey of the Literature, Trends and Prospects,The American Statistician 27(1), 15-17.
[15] Lawless, J.F. (1982),Statistical Models and Methods for Lifetime Data, New York, Wiley. · Zbl 0541.62081
[16] Lieblein, J. and Zelen, M. (1956), Statistical Investigation of the Fatigue Life of Deep-Groove Ball Bearings,Journal of Research of the National Bureau of Standards 47, 273-316.
[17] Lochner, H.R., When and How to Use the Weibull Distribution, Research Report.
[18] McCool, J.I. (1970), Inference on Weibull Percentiles and Shape Parameters from Maximum Likelihood Estimates,IEEE Transactions on Reliability 19, 2-9.
[19] McCool, J.I. (1974), Inferential techniques for Weibull populations,Aerospace Research Laboratories Report ARL TR 74-0180, Wright-Patterson AFB, Ohio. · Zbl 0305.62066
[20] Moore, R.E. (1979),Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, Philadelphia. · Zbl 0417.65022
[21] Panchang, V.G. and Gupta, R.C. (1989), On the Determination of Three-Parameter Weibull MLE’s,Communications in Statistics and Computation 18(3), 1037-1057. · Zbl 0695.62073
[22] Petrauskas, A. and Aagaard, P.M. (1971), Extrapolation of Historical Storm Data for Estimating Design Wave Heights,Society of Petroleum Engineers Journal 11, 23-37.
[23] Piyavskii, S.A. (1967), ?An Algorithm for Finding the Absolute Minimum of a Function?,Theory of Optimal Solutions 2, Kiev, IK AN USSR (in Russian), pp. 13-24.
[24] Piyavskii, S.A. (1972), An Algorithm for Finding the Absolute Extremum of a Function,USSR Computational Mathematics and Mathematical Physics 12, 57-67. · Zbl 0282.65052
[25] Ratschek, H. and Rokne, J. (1984),Computer Methods for the Range of Functions, Ellis Horwood Series Mathematics and its Applications, Wiley, New York. · Zbl 0584.65019
[26] Ratschek, H. and Rokne, J. (1988),New Computer Methods for Global Optimization, Ellis Horwood, Chichester. · Zbl 0648.65049
[27] Rockette, H., Antle, C. and Klimko, L.A. (1974), Maximum Likelihood Estimation with the Weibull Model,Journal of the American Statistical Association 69(345), 246-249. · Zbl 0283.62033
[28] Wingo, D.R. (1972), Maximum Likelihood Estimation of the Parameters of the Weibull Distribution by Modified Quasilinearization,IEEE Transactions on Reliability 21, 89-93.
[29] Wingo, D.R. (1973), Solution of the Three-Parameter Weibull Equations by Constrained Modified Quasilinearization (Progressively Censored Samples),IEEE Transactions on Reliability 22(2), 96-102.
[30] Zanakis, S.H. (1977), Computational Experience with Some Nonlinear Optimization Algorithms in Deriving Maximum Likelihood Estimates for the Three-Parameter Weibull Distribution, inAlgorithmic Methods in Probability, M.F. Neuts (ed.), TIMS Studies in Management Sciences, North-Holland7, 63-77. · Zbl 0396.62075
[31] Zanakis, S.H. (1979), Extended Pattern Search with Transformations for the Three-Parameter Weibull MLE Problem,Management Science 25(11), 1149-1161. · Zbl 0474.62091
[32] Zanakis, S.H. and Kyparisis, J. (1986), A Review of Maximum Likelihood Estimation Methods for the Three-Parameter Weibull Distribution,Journal of Statistical Computation and Simulation 25, 53-73. · Zbl 0596.62028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.