Preservation properties of a homogeneous Poisson process stopped at an independent random time

To study the ageing and stochastic properties of lifetime random variables, several classes of lifetime distributions have been defined in reliability literature. In this context, an interesting problem is then to investigate the closure properties of such classes under different operators. In this paper, we study the preservation properties of classes of increasing mean inactivity time (IMIT), increasing variance inactivity time (IVIT), and some other recently introduced classes, such as NRBU, NRBUE, and NBRUrh, under a Poisson process stopped at a random time. We also analyze the preservation properties of Poisson shock models for the above-mentioned ageing classes.