Evaluation of the Robustness of MLE Method for Selecting the Best Fitting Lifetime Distribution

Abstract

Warranty data and other time-to-failure data is frequently analyzed through the estimation of the parameters of one or more lifetime distributions including: two-parameter Weibull, three-parameter Weibull, Lognormal, Logistic, Gumbel, Gamma. The form of some lifetime distributions is derived from the nature of the failure process, e.g., the Gumbel is used to model the maximum of a number of samples from the same distribution. Other lifetime distributions, e.g., Weibull can be used to fit a wide variety of failure mechanisms. When the form of the lifetime distribution is not specified by the failure mechanism, and there is no historical precedent demonstrating that a particular family of distributions is a good fit for the data in question, goodness of fit methods can be used to choose the form of the lifetime distribution that best fits the available data. The lifetime distribution chosen may have a large impact on the inferences drawn from the data, particularly when the subject of interest is future reliability or reliability projected beyond the range of the currently available data. Maximum Likelihood is one of the most popular distribution selection methods. The robustness of Maximum Likelihood for choosing the ‘best’ lifetime distribution, and the impact of choosing the wrong distribution was evaluated by using Monte Carlo simulation to generate samples from four specified lifetime distributions. The Reliasoft® Weibull++ software and SAS were used to estimate the best fitting distribution and to estimate the B50, age at 50% failure, B90, the age at 90% failure. Results of the research include the proportion of Monte Carlo runs that the specified lifetime distribution was correctly chosen as the best fitting distribution, and comparison of the accuracy of estimates of B50 and B90 when calculated from the best fit distribution as compared to the B50 and B90 when calculated with estimated parameters for the Actual distribution. In this study, the Lognormal distribution was usually correctly selected while the Weibull and the Logistic distribution were frequently misidentified as another lifetime distribution. Suggestions for addressing the practical implications of misidentified lifetime distributions are discussed.