Maximum likelihood estimation for the poly-Weibull distribution

Abstract

The Weibull distribution has long been a popular choice for modeling lifetime data of various mechanical and biological phenomena when the associated hazard rate function is constant or monotone increasing or decreasing. However, nonmonotone hazard functions are common in reliability and survivability contexts where a system may undergo an initial “burn-in” prior to periods of useful life and eventual wear out. In these scenarios, the Weibull can only model a portion of the “bathtub” curve but is incapable of adequately modeling the entire failure process. Several modifications to the standard two-parameter Weibull distribution have therefore been introduced in the literature to effectively model and analyze lifetime data where the hazard rate function is bathtub-shaped. The performance of each modified distribution is typically assessed by its ability to fit a reference data set that is known to have a bathtub-shaped hazard rate function. The current article compares the performance of two recent contributions in this area to that of the poly-Weibull distribution with respect to several goodness-of-fit measures. In addition, numerical and analytical procedures are developed for obtaining the maximum likelihood parameter estimates, standard errors, and an equation to determine the moments for the generalized poly-Weibull distribution with arbitrary number of terms. Our results show that both the bi-Weibull and tri-Weibull distributions fit the reference data set better than the current best-fit models.