Optimal hazard models based on partial information

The proportional hazards (PH), mixture hazards (MH), proportional reversed hazards (PRH), and mixture reversed hazards (MRH) models are widely used in various applications in many fields, but their optimality properties remain unknown. We represent these important reliability models as general escort models and derive them as solutions to several information theoretic formulations. Thus far, the escort models are defined in physics by the normalized powers of one or product of two probability mass or density functions and have been derived in statistics and physics as the solutions to different information formulations. The general escort models introduced in this paper include the escorts of densities, as well as the escorts of survival functions and cumulative distribution functions which represent the hazards models. Moreover, we show that the MH and MRH models are also optimal according to formulations in terms of the mean variation distance. Additional results explore reliability properties of the escort of two densities. A notable property is that the escort of two densities with non-constant hazard rates can be a constant hazard rate model. Another result characterizes the PH model in terms of the survival function of the escort of two densities. Comparisons of the MH, escort of two densities, and the mixture of two distributions are illustrated.