A note on the proportional hazards model with discontinuous data

Cox's proportional hazards (PH) model is applicable to continuous random (response) variables as well as to discontinuous ones. We make two remarks on the PH models for discontinuous random (response) variables. (1) In general, the proportional hazards relation can only occur in the interior of the support of the two relevant random variables, instead on the whole support, as stated in the standard textbooks. (2) The PH model is not the same as a proportional cumulative hazards model (or a Lehmann family) unless the random variables are continuous. These two models are mistaken to be the same in several papers on Cox's regression model in the literature.