A note on the connection between some classical mortality laws and proportional frailty

We provide a simple frailty argument that produces the Gompertz-Makeham mortality law as the population hazard rate under the assumption of proportional frailty given a common exponential hazard rate. Further, based on a slight generalisation of the result for the Gompertz-Makeham law the connection to Perks and Beard's mortality laws is discussed. Moreover, we give conditions for which functional forms of the baseline hazard that will yield proper frailty distributions given we want to retrieve a certain overall population hazard within the proportional frailty framework.