Estimation of Weibull distribution parameters based on sequences of minimal repairs

• Autorzy: Malinowski Jacek

Identyfikatory

DOI

10.26408/srsp-2022-11

• Konferencja: 16th Summer Safety & Reliability Seminars - SSARS 2022, 4-11 September 2022, Ciechocinek, Poland
• Języki publikacji: EN

Abstrakty

EN

A new method of estimating the scale and shape parameters of the Weibull distribution is presented. According to this method, a Weibull distributed time-to-failure (TTF) of a test item is measured m times. It undergoes a minimal repair after each of the first m-1 failures, and is put out of use after the m-th failure. This procedure is repeated n times. Based on m TTFs of one test item, which are neither independent nor identically distributed (IID), the maximum likelihood estimators (MLE) of the scale and shape parameters, called n m-sample estimators, are obtained. The accuracy of the m-sample estimators is low, however, it can be improved by using the mean values of their n IID realizations as more precise estimators. The latter are called n·m-sample estimators, have the same biases as the respective m-sample ones, but their variances are n times smaller. Interestingly enough, the n·m-sample estimators of the scale and shape parameters, as well as their biases, are given by relatively simple explicit formulas. This is somewhat unexpected in view of the fact that the standard MLE of the shape parameter, based on IID TTFs of non-repairable test items, is obtained from an equation that cannot be solved analytically.

Słowa kluczowe

EN

Weibull distribution; minimal repair; maximum likelihood estimation; gamma function; Euler constant

• Wydawca: Gdynia Maritime University ; Polskie Towarzystwo Bezpieczeństwa i Niezawodności
• Czasopismo: Safety and Reliability of Systems and Processes
• Rocznik: 2022
• Tom: Summer Safety and Reliability Seminar 2022
• Strony: 163--172
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Malinowski Jacek

• Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

• https://orcid.org/0000-0002-4413-1868

Bibliografia

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• Chikr el-Mezouar, Z. 2010. Estimation of the shape, location and scale parameters of the Weibull distribution. Reliability: Theory & Applications (Electronic Journal of International Group on Reliability) 5(4), 3-40.
• Dodson, B. 2006. The Weibull Analysis Handbook. Second Edition. American Society for Quality.
• Evans, J.W., Kretschmann, D.E. & Green, D.W. 2019. Procedures for estimation of Weibull Parameters. United States Department of Agriculture, Forest Service, Forest Products Laboratory. General Technical Report FPL-GTR-264.
• Feller, W. 1971. An Introduction to Probability Theory and its Applications. John Wiley & Sons, New York.
• Lei, Y. 2008. Evaluation of three methods for estimating the Weibull distribution parameters of Chinese pine. Journal of Forest Science 54(12), 566-571.
• Wu, Y., Xie, H., Chiang, J.-Y., Peng, G. & Qin, Y. 2021. Parameter estimation and applications of the Weibull distribution for strength data of glass fiber. Hindawi. Mathematical Problems in Engineering 2021, article ID 9175170.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).