A bivariate Gompertz–Makeham life distribution

In the context of actuarial science, Gompertz (1825) utilized a differential equation to derive the life distribution that carries his name. Subsequently, De Morgan (1860), Woolhouse (1863), and Kaminsky (1983) derived the Gompertz distribution from functional equations. This paper focuses on bivariate versions of Kaminsky’s functional equation. A limiting version yields the bivariate exponential distribution of Marshall and Olkin (1967).