Mixed Weibull distributions in reliability engineering: Statistical models for the lifetime of units with multiple modes of failure.

Abstract

The finite mixed Weibull distribution is an appropriate distribution in modeling the lifetime of the units having more than one possible failure cause. Due to the lack of a systematic statistical procedure of fitting the distribution to a data set, it has not been widely used in lifetime data analyses. Many areas on this subject have been studied in this research. The following are the findings and contributions. Through a change of variable, 5 parameters in a two Weibull mixture can be reduced to 3. A parameter'vector (p₁, η, β) defines a family of two-Weibull mixtures which have common characteristics. Numerous probability plots are investigated on Weibull probability paper (WPP). For a given p₁ the η-β plane is partitioned into seven regions which are labeled by A through F and S. The Region S represents the two Weibull mixtures whose cdf curves are very close to a straight line. The Regions A through F represent six typical shapes of the cdf curves on WPP, respectively. The two-Weibull mixtures in one region have similar characteristics. Three important features of the two-Weibull mixture with well separated subpopulations are proved. Two existing methods for the graphical estimation of the parameters are discussed, and one is recommended over the other. The EM algorithm is successfully applied to solve the MLE for mixed Weibull distributions when m, the number of subpopulations in a mixture is known. The algorithms for complete, censored, grouped and suspended samples with non-postmortem and postmortem failures are developed accordingly. The developed algorithms are powerful, efficient and they are insensitive to the initial guesses. Extensive Monte Carlo simulations are performed. The distributions of the MLE of the parameters and of the reliability of a two Weibull mixture are studied. The MLEs of the parameters are sensitive to the degree of separation of the two subpopulation pdfs, but the MLE of the reliability is not. The generalized likelihood ratio (GLR) test is used to determine m. Under H₀: m=1 and H₁: m=m₁>1, ζ, the GLR is independent of the parameters in the distribution of H₀. The distributions of ζ or -21n(ζ) with n=50, 100 and 150 are obtained through Monte Carlo simulations. Compared with the chi-square distribution, they fall in between x²(4) and x²(6), and they are very close to x²(5). A FORTRAN computer program is developed to conduct simulation of the GLR test for 1 ≤ m₀ < m₁ ≤ 5.