×

Estimation of the reliability parameter for three-parameter Weibull models. (English) Zbl 1481.62093

Summary: For \(X\) and \(Y\) independent three-parameter Weibull random variables, the estimation of the reliability parameter \(\delta = P(Y < X)\) is important in applications of failure times and stress-strength situations in industry, medicine, extreme value theory, hydrology, and environmetrics. Estimation problems arise when the likelihood function is not defined properly, since the three-parameter Weibull distribution is non regular and its density has singularities. Here, a correct likelihood for the three-parameter Weibull case is proposed for the first time, in the spirit of the original definition of likelihood. It takes into account the occurrence of the smallest observation with respect to the threshold parameter, as well as the fact that all measuring instruments necessarily have a finite precision. Possible repeated observations are immediately explained. When the Weibull distributions of \(X\) and \(Y\) have common threshold and shape parameters, the profile likelihood can be used for inferences about \(\delta\). For the case of all three Weibull parameters unknown and arbitrary, inferences about \(\delta\) are obtained via a novel Bootstrap approach. An example previously analyzed under alternative inferential approaches is presented to illustrate the convenience of the proposal.

MSC:

62N05 Reliability and life testing
62F10 Point estimation
62F40 Bootstrap, jackknife and other resampling methods
60E05 Probability distributions: general theory

Software:

bootstrap
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Elmahdy, E. E.; Aboutahoun, A. W., A new approach for parameter estimation of finite Weibull mixture distributions for reliability modeling, Appl. Math. Model., 37, 4, 1800-1810 (2013) · Zbl 1349.90249
[2] Elmahdy, E. E., A new approach for Weibull modeling for reliability life data analysis, Appl. Math. Comput., 250, 708-720 (2015) · Zbl 1328.62573
[3] Rinne, H., The Weibull Distribution: A Handbook (2009), Chapman & Hall/CRC, Boca Raton, FL · Zbl 1270.62040
[4] Hirose, H.; Lai, T. L., Inference from grouped data in three-parameter Weibull models with applications to breakdown-voltage experiments, Technometrics, 39, 2, 199-210 (1997) · Zbl 0889.62088
[5] Green, E. J.; Roesch Jr, F. A.; Smith, A. F.; Strawderman, W. E., Bayesian estimation for the three-parameter weibull distribution with tree diameter data, Biometrics, 50, 254-269 (1994) · Zbl 0825.62401
[6] Khan, H. M.R.; Albatineh, A.; Alshahrani, S.; Jenkins, N.; Ahmed, N. U., Sensitivity analysis of predictive modeling for responses from the three-parameter Weibull model with a follow-up doubly censored sample of cancer patients, Comput. Stat. Data Anal., 55, 12, 3093-3103 (2011) · Zbl 1271.62220
[7] Datsiou, K. C.; Overend, M., Weibull parameter estimation and goodness-of-fit for glass strength data, Struct. Safety, 73, 29-41 (2018)
[8] Deng, B.; Jiang, D.; Gong, J., Is a three-parameter Weibull function really necessary for the characterization of the statistical variation of the strength of brittle ceramics?, J. Eur. Ceramic Soc., 38, 4, 2234-2242 (2018)
[9] Sanni, S.; Chukwu, W., An economic order quantity model for items with three-parameter Weibull distribution deterioration, ramp-type demand and shortages, Appl. Math. Model., 37, 23, 9698-9706 (2013) · Zbl 1427.90028
[10] Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall: I. theoretical investigation, Hydrol. Sci. J., 49, 4, 575-590 (2004)
[11] Silva, H.; Peiris, T., Statistical modeling of weekly rainfall: a case study in colombo city in Sri Lanka, Proceedings of the Engineering Research Conference (MERCon), Moratuwa, 241-246 (2017), IEEE
[12] Bolívar-Cimé, A.; Díaz-Francés, E.; Ortega, J., Optimality of profile likelihood intervals for quantiles of extreme value distributions: application to environmental disasters, Hydrol. Sci. J., 60, 3-4, 651-670 (2015)
[13] Kotz, S.; Lumelskii, Y.; Pensky, M., The Stress-Strength Model and its Generalizations: Theory and Applications (2003), World Scientific, New Jersey · Zbl 1017.62100
[14] Cousineau, D., Fitting the three-parameter Weibull distribution: Review and evaluation of existing and new methods, IEEE Trans. Dielectr. Electr. Insulat., 16, 1, 281-288 (2009)
[15] Cheng, R. C.H.; Iles, T. C., Corrected maximum likelihood in non-regular problems, J. R. Stat. Soc. Ser B., 49, 1, 95-101 (1987) · Zbl 0615.62028
[16] Lawless, J. F., Statistical Models and Methods for Lifetime Data (2003), Wiley-Interscience: Wiley-Interscience New York · Zbl 1015.62093
[17] Montoya, J. A.; Díaz-Francés, E.; Sprott, D. A., On a criticism of the profile likelihood function, Stat. Pap., 50, 1, 195-202 (2009) · Zbl 1312.62010
[18] Liu, S.; Wu, H.; Meeker, W. Q., Understanding and addressing the unbounded “likelihood” problem, Am. Stat., 69, 3, 191-200 (2015) · Zbl 07671730
[19] Harper, W.; Eschenbach, T.; James, T., Concerns about maximum likelihood estimation for the three-parameter Weibull distribution: case study of statistical software, Am. Stat., 65, 44-54 (2011)
[20] Kundu, D.; Raqab, M., Estimation of \(R = P(Y < X)\) for three-parameter Weibull distribution, Stat. Probab. Lett., 79, 1839-1846 (2009) · Zbl 1169.62012
[21] Smith, R. L., Maximum likelihood estimation in a class of nonregular cases, Biometrika, 72, 1, 67-90 (1985) · Zbl 0583.62026
[22] Fisher, R. A., On the mathematical foundations of theoretical statistics, Phil. Trans. R. Soc. Lond., 222, 594-604, 309-368 (1922) · JFM 48.1280.02
[23] Barnard, G. A., The use of the likelihood function in statistical practice, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1, 27-40 (1967), Berkeley, Calif. · Zbl 0223.62005
[24] Lindsey, J. K., Some statistical heresies, J. R. Stat. Soc. Ser. D (Stat.), 48, 1-40 (1999)
[25] Kalbfleisch, J. G., Probability and Statistical Inference, 2 (1985), Springer-Verlag New York · Zbl 0589.62002
[26] Brown, G. G.; Rutemiller, H. C., Evaluation of pr{\(x\) ≥ \(y\)} when both x and y are from three-parameter Weibull distributions, IEEE Trans. Reliab., 22, 2, 78-82 (1973)
[27] Figueroa, G., Las funciones de verosimilitud discretizada y restringida perfil en la Inferencia Científica (2012), Department of Mathematics, University of Sonora, Ph.D. thesis
[28] Barndorff-Nielsen, O. E.; Cox, D. R., Inference and asymptotics, Monographs on Statistics and Applied Probability (1994), Boca Raton: Chapman & Hall/CRC · Zbl 0826.62004
[29] Pawitan, Y., In All Likelihood: Statistical Modelling and Inference Using Likelihood (2001), Oxford: Clarendon Press · Zbl 1013.62001
[30] Sprott, D. A., Statistical Inference in Science (2000), Springer-Verlag New York, Inc. · Zbl 0955.62006
[31] Serfling, R. J., Approximation Theorems of Mathematical Statistics (2002), John Wiley and Sons, New York · Zbl 1001.62005
[32] Efron, B., Bootstrap methods: another look at the jackknife, Ann. Stat., 7, 1, 1-26 (1979) · Zbl 0406.62024
[33] Efron, B.; Tibshirani, R. J., An Introduction to the Bootstrap (1993), Chapman & Hall/CRC · Zbl 0835.62038
[34] Bader, M.; Priest, A., Statistical aspects of fiber and bundle strength in hybrid composites, (Hayashi, T.; Kawata, S.; Umekawa, S., Proceeding of the Progress in Science and Engineering Composites, ICCM-IV, Tokyo (1982)), 1129-1136
[35] Kundu, D.; Gupta, R. D., Estimation of \(P[Y < X]\) for Weibull distributions, IEEE Trans. Reliab., 55, 2, 270-280 (2006)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.