A note on the existence of the location parameter estimate of the three-parameter Weibull model using the Weibull plot. (English) Zbl 1428.62107

Summary: The Weibull model is one of the widely used distributions in reliability engineering. For the parameter estimation of the Weibull model, there are several existing methods. The method of the maximum likelihood estimation among others is preferred because of its attractive statistical properties. However, for the case of the three-parameter Weibull model, the method of the maximum likelihood estimation has several drawbacks. To avoid the drawbacks, the method using the sample correlation from the Weibull plot is recently suggested. In this paper, we provide the justification for using this new method by showing that the location estimate of the three-parameter Weibull model exists in a bounded interval.


62F10 Point estimation
62N05 Reliability and life testing


Full Text: DOI


[1] Barnard, G. A., The use of the likelihood function in statistical practice, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press
[2] Smith, R. L.; Naylor, J. C., A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution, Journal of the Royal Statistical Society: Series C (Applied Statistics), 36, 3, 358-369, (1987)
[3] Gumbel, E., Statistical forecast of droughts, Hydrological Sciences Journal, 8, 1, 5-23, (1963)
[4] Vogel, R. M.; Kroll, C. N., Low-flow frequency analysis using probability-plot correlation coefficients, Journal of Water Resources Planning and Management, 115, 3, 338-357, (1989)
[5] Cheng, R. C.; Amin, N. A., Estimating parameters in continuous univariate distributions with a shifted origin, Journal of the Royal Statistical Society Series B (Methodological), 45, 3, 394-403, (1983) · Zbl 0528.62017
[6] Cheng, R. C.; Iles, T. C., Embedded models in three-parameter distributions and their estimation, Journal of the Royal Statistical Society Series B (Methodological), 52, 1, 135-149, (1990) · Zbl 0699.62026
[7] Liu, S.; Wu, H.; Meeker, W. Q., Understanding and addressing the unbounded “likelihood” problem, The American Statistician, 69, 3, 191-200, (2015)
[8] Huzurbazar, V. S., The likelihood equation, consistency and the maxima of the likelihood function, Annals of Eugenics, 14, 185-200, (1948) · Zbl 0033.07703
[9] Sirvanci, M.; Yang, G., Estimation of the weibull parameters under type I censoring, Journal of the American Statistical Association, 79, 385, 183-187, (1984) · Zbl 0533.62029
[10] Park, C., Weibullness test and parameter estimation of the three-parameter weibull model using the sample correlation coefficient, International Journal of Industrial Engineering : Theory, Applications and Practice, 24, 4, 376-391, (2017)
[11] Nelson, W., Applied Life Data Analysis, (1982), New York, NY, USA: John Wiley and Sons, New York, NY, USA · Zbl 0579.62089
[12] Blom, G., Statistical Estimates and Transformed Beta-Variables, (1958), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0086.34501
[13] Wilk, M. B.; Gnanadesikan, R., Probability plotting methods for the analysis of data, Biometrika, 55, 1, 1-17, (1968)
[14] Park, C., Weibullness: Goodness-Of-Fit Test for Weibull Distribution (R Package), (2018)
[15] Reliability HotWire, Reliability Basics – Location Parameter of The Weibull Distribution, 15, (May 2002)
[16] Wong, K. L., When reliable means failure free, Quality and Reliability Engineering International, 3, 3, 145-145, (1987)
[17] Clech, J.-P. M.; Noctor, D. M.; Manock, J. C.; Lynott, G. W.; Bader, F. E., Surface mount assembly failure statistics and failure free time, Proceedings of the 1994 IEEE 44th Electronic Components & Technology Conference
[18] Drapella, A., An improved failure-free time estimation method, Quality and Reliability Engineering International, 15, 3, 235-238, (1999)
[19] Mitchell, D.; Zahn, B.; Carson, F., Board level thermal cycle reliability and solder joint fatigue life predictions of multiple Stacked Die chip scale package configurations, Proceedings of the 54th Electronic Components and Technology Conference
[20] Lam, S.-W.; Halim, T.; Muthusamy, K., Models with failure-free life—applied review and extensions, IEEE Transactions on Device and Materials Reliability, 10, 2, 263-270, (2010)
[21] de Bruijn, N. G., Asymptotic Methods in Analysis, (1981), New York, NY, USA: Dover Publications, New York, NY, USA · Zbl 0556.41021
[22] Apostol, T. M., Mathematical Analysis, (1974), Reading, Mass, USA: Addison-Wesley Publication, Reading, Mass, USA
[23] Bartle, R. G.; Sherbert, D. R., Introduction to Real Analysis, (1982), New York, NY, USA: John Wiley and Sons, New York, NY, USA · Zbl 0494.26002
[24] Hwang, T. Y.; Hu, C. Y., The best lower bound of sample correlation coefficient with ordered restriction, Statistics and Probability Letters, 19, 3, 195-198, (1994) · Zbl 0791.62062
[25] Farnum, N. R.; Booth, P., Uniqueness of maximum likelihood estimators of the 2-parameter weibull distribution, IEEE Transactions on Reliability, 46, 4, 523-525, (1997)
[26] Bilikam, J. E., k-sample maximum likelihood ratio test for change of weibull shape parameter, IEEE Transactions on Reliability, R-28, 1, 47-50, (1979) · Zbl 0395.62042
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