Ordering properties of the smallest and largest lifetimes in Gompertz–Makeham model

The Gompertz–Makeham (GM) distribution, which is used commonly to represent lifetimes based on laws of mortality, is one of the most popular choices for mortality modeling in the field of actuarial science. This paper investigates ordering properties of the smallest and largest lifetimes arising from two sets of heterogeneous groups of insurees following respective GM distributions. Some sufficient conditions are provided for comparing the smallest and largest lifetimes from two sets of dependent variables in the sense of usual stochastic ordering. Comparison results on the smallest lifetimes, in the sense of hazard rate ordering and ageing faster ordering, are established for two groups of heterogeneous independent lifetimes. Under a similar set-up, no reversed hazard rate ordering is shown to exist between the largest lifetimes with the use of a counter-example. Finally, sufficient conditions are presented for comparing two sets of independent heterogeneous lifetimes under random shocks by means of usual stochastic ordering. Such comparisons for the smallest lifetimes are also carried out in terms of hazard rate ordering.