Fitting the Weibull and lognormal log-linear models to accelerated life test data

Abstract

Accelerated life tests, in which more than one stress is often involved, have become widely used in today's industries to obtain the time-to-failure and reliability information at normal use conditions. Tests are conducted at higher than normal levels of stresses to shorten the test duration. A physical-statistical model is needed to extrapolate the results from test conditions to usage conditions. The generalized Weibull and lognormal log-linear models, as two general ALT families, cover almost all ALT models which are current in use in reliability engineering for this purpose. However, the development of multiple-stress ALTs has been hindered by the difficulty of performing adequate and satisfactory model fitting. This study presents an extensive research on both point and interval estimates of model parameters. The maximum likelihood estimates (MLE), as the first choice of the point estimate, have preferable statistical properties and well-developed theories. Due to complication of the models and data patterns, a robust and efficient algorithm is essential to successful implementation of the ML estimation. Unfortunately, the current methods get impractical, and no effective and practical approach has been developed yet for the generalized Weibull and lognormal log-linear models. A new approach to obtain ML point estimators of the parameters for both models, which takes advantage of generalized linear model (GLM), has been proposed and extensively studied in this research. The algorithm is generally numerically stable and easily programmed. The superiority is that it does not depend much on the starting values. This proposed method might generate a long-standing method to obtain the MLE for the ALT and other models which have two sets of unknown parameters, one in the mean function and other in the variance function. The likelihood ration confidence intervals have been concluded generally to be the best among the available approximate confidence methods, based on recent researches. The LR confidence bound method is successfully applied to calculate the confidence limits on the reliability under the use conditions in this study. Furthermore, the study has established a general method to calculate the LR ratio confidence limits on a function of unknown parameters. The procedures of point and interval estimates have been developed and their virtues have been demonstrated with several numerical examples of actual accelerated life test data.