Estimation of Weibull distribution parameters based on sequences of minimal repairs

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.

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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).