Constraints for generalized mixtures of Weibull distributions with a common shape parameter

Weibull mixtures have been widely used in many applications, and they can be generalized by allowing negative mixing weights. In this note, we investigate constraints on the mixing weights and parameters of components under which the generalized mixture of Weibull distributions is a valid probability model, including the cases of exponential or Rayleigh components. In addition, characterizations are shown for generalized mixtures of three or fewer Weibull components. We use an example of a two-fold Weibull mixture model where negative weights are estimated, and the constraints are illustrated with a graphical example of three components.