### Parameter estimation for mixed-Weibull distribution

D.B. Kececioglu, Wendai Wang
Annual Reliability and Maintainability Symposium. 1998 Proceedings. International Symposium on Product Quality and Integrity
& CONCL USIONS In reliability engineering, it is known that electrical and mechanical equipment usually have more than one failure mode or cause. It has been recognized for more than three decades that the mixed Weibull distribution is an appropriate distribution to use in modeling the lifetimes of the units that have more than one failure cause. However, due to the lack of a systematic statistical procedure for fitting an appropriate distribution to such a mixed data set, it has not been
more » ... used. A mixed Weibull distribution represents a population that consists of several Weibull subpopulations. In this paper, a new approach is developed to estimate the mixed-Weibull distribution's parameters. At first, the population sample data are split into subpopulation data sets over the whole test duration by using the posterior belonging probability of each observation to each subpopulation. Then, with the new concepts of Fracture Failure and Mean Order Number, the proposed approach combines the Least-Squares method with Bayes' Theorem, takes advantage of the parameter estimation for single Weibull distribution to each derived subgroup data set, and estimates the parameters of each subpopulation. The proposed approach can also be applied for complete, censored, and grouped data samples. Its superiority is particularly significant when the sample size is relatively small and for the case in which the subpopulations are well mixed. A numerical example is given to compare the proposed method with the conventional plotting method of subpopulation separation. It turns out that the proposed method yields more accurate parameter estimates. NOTA TION f (4 Probability density hnction, p d ! of a mixed population x (4 Probability density function, pdJ; of jth subpopulation, j = 1 , 2 4 , vj P, 9 Weibull shape and scale parameters of \$ (t) Mixing weight for Subpopulation 1 and 2, p E (0, 11, p+q = 1 Posterior belonging probability Mean Order Number Median Rank loge {-loge[ 1 -MRj (till 1 loge (ti> Correlation coefficients, p = p1 + pz