Higher Order Equilibrium Life Distributions

In this work, we define the higher order old age distribution, also known as the higher order equilibrium distribution, and establish some properties pertaining to its distribution, moments, moment generating function, hazard rate, and mean residual life. When a certain higher order old age distribution belongs to one of several well-known aging families, we obtain some moment inequalities of the underlining life. These results are useful in reliability studies, as well as in renewal theory, actuarial science, and economics.