Weibull and inverse Weibull mixture models allowing negative weights

This article presents the finding that when components of a system follow the Weibull or inverse Weibull distribution with a common shape parameter, then the system can be represented by a Weibull or inverse Weibull mixture model allowing negative weights. We also use an example to illustrate that the proposed mixture model can be used to approximate the reliability behaviours of the consecutive-k-out-of-n systems. The example also shows data analysis procedures when the parameters of the component life distributions are either known or unknown.