Estimation of the threshold parameter of a wear-out failure period in the case of a three-parameter Weibull distribution

We aimed to estimate the threshold parameter for the wear-out failure period of a three-parameter Weibull distribution. In this paper, we propose the minimum-variance linear unbiased estimator based on order statistics, which is denoted by 'TBEST'. In Section 2, we verify the validity of TBEST by comparing it with the existing threshold parameter estimator, based on simulation studies of bias and mean squared error (MSE). Our results show that TBEST requires all order statistics, except for the case of an exponential distribution, in which TBEST is reduced to an unbiased estimator based on the smallest observation only. In Section 3, by simulation studies, we compare TBEST and other estimators known to have good performances. In the simulation results, bias and MSE of TBEST were the smallest in most cases. We also show a numerical example to measure fatigue lives in hours of ten bearings from McCool (1974).