A family of bivariate failure time distributions with proportional crude and conditional hazards
A hazard regression model is formulated from consideration of the crude hazard function in a competing risks setup and the conditional hazard function in a bivariate failure time model.Specifically, we model the bivariate survival function with a proportional hazards model for the first failure and a conditional proportional hazards model for the second failure given the time and type of the first failure.A family of distributions satisfying the model constraints is identified, and some special cases are presented.Some properties of the bivariate distribution are derived, including necessary and sufficient conditions for the existence and uniqueness of the distribution.Identifiability of the parameters of the bivariate distribution is discussed, as well as necessary and sufficient conditions for statistical independence of the two survival times conditional on the realized random covariate vector.