MAXIMUM-LIKELIHOOD-ESTIMATES, FROM CENSORED-DATA, FOR MIXED-WEIBULL DISTRIBUTIONS

The mixed-Weibull distribution provides a good model for the lives of electrical and mechanical components (or systems) when the failure of the components (or systems) is caused by more than one failure mode. Due to the lack of an efficient parameter estimation method, the mixed-Weibull model has not been used as widely by reliability practitioners as the single-population Weibull distribution. This paper presents a new algorithm for estimating the parameters of mixed-Weibull distributions from censored data. The algorithm follows the principle of the MLE (maximum likelihood estimate) through the EM (expectation and maximization) algorithm, and it is derived for both postmortem and non-postmortem time-to-failure data. The following conclusions are drawn: 1) The concept of the EM algorithm is easy to understand and apply (only elementary statistics and calculus are required). 2) The log-likelihood function can not decrease after an EM sequence; this important feature was observed in all of the numerical calculations. 3) The MLEs of the non-postmortem data were obtained successfully for mixed-Weibull distributions with up to 14 parameters in a 5-subpopulation, mixed-Weibull distribution. This has not been seen, numerically, in the literature even for 3-subpopulation Weibull mixtures. We believe that there are no further difficulties in obtaining the MLEs of mixed-Weibull distributions even with more than 5 subpopulations. 4) The algorithm for the MLEs of postmortem data is a special case of our algorithm for the MLEs of non-postmortem data. 5) Numerical examples indicate that some of the log-likelihood functions of the mixed-Weibull distributions have multiple local maxima; therefore, the algorithm should start at several initial guesses of the parameters set. The searching of the largest local maximum can stop when a good fit has been found. 6) The EM algorithm is very efficient. On the average for 2-Weibull mixtures with a sample size of 200, the CPU time (on VAX 8650) is 0.13 seconds per iteration. The number of iterations depends on the characteristics of the mixture. The number of iterations is small if the subpopulations in the mixture are well separated. Generally, the algorithm is not sensitive to the initial guesses of the parameters.