Parameter Estimation of Limited Failure Population Model With a Weibull Underlying Distribution

For many data-driven reliability problems, the population is not homogeneous; i.e., its statistics are not described by a unimodal distribution. Also, the interval of observation may not be long enough to capture the failure statistics. A limited failure population (LFP) consists of two subpopulations, a defective and a nondefective one, with well-separated modes of the two underlying distributions. In reliability and warranty forecasting applications, the estimation of the number of defective units and the estimation of the parameters of the underlying distribution are very important. Among various estimation methods, the maximum likelihood estimation (MLE) approach is the most widely used. Its likelihood function, however, is often incomplete, resulting in an erroneous statistical inference. In this paper, we estimate the parameters of a LFP analytically using a rational function fitting (RFF) method based on the Weibull probability plot (WPP) of observed data. We also introduce a censoring factor (CF) to assess how sufficient the number of collected data is for statistical inference. The proposed RFF method is compared with existing MLE approaches using simulated data and data related to automotive warranty forecasting.