Reliability approximation using finite Weibull mixture distributions

Abstract

The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that the reliability of an arbitrary system can be approximated well by a finite Weibull mixture with positive component weights only, without knowing the structure of the system, on condition that the unknown parameters of the mixture can be estimated. To support the main idea, five examples are presented. In order to estimate the unknown component parameters and the component weights of the Weibull mixture, some of the already existing methods are applied and the EM algorithm for the m-fold Weibull mixture is derived. The fitted distributions obtained by different methods are compared to the empirical ones by calculating the AIC and δC values. It can be concluded that the suggested Weibull mixture with an arbitrary but finite number of components is suitable for lifetime data approximation. For parameter estimation the combination of the alternative and EM algorithm is suggested.