Reliability properties of generalized mixtures of Weibull distributions with a common shape parameter

Weibull mixtures have been considered in many applied problems, and they have also been generalized by allowing negative mixing weights. In this paper, we study the classification of the aging properties of generalized mixtures of two or three Weibull distributions in terms of the mixing weights, scale parameters and a common shape parameter, which extends the cases of exponential or Rayleigh distributions. We apply these general results to classify the aging properties of the minimum and maximum lifetimes of two-component systems whose component lifetimes follow one of the known bivariate Weibull distributions.