Classes of life distributions based on the moment-generating-function order are investigated in this paper. It is shown firstly that the class ℳ is closed under both convex linear combination and geometric compounding. Secondly, the class NBUmg (new better than used in the moment-generating-function order) is proved to be closed under increasing star-shaped transformations. Finally, the interplay between the stochastic comparison of the excess lifetime of a renewal process and the NBUmg interarrivals is studied.

We introduce a new class of lifetime distributions exhibiting a notion of positive ageing, called the ℳ-class, which is strongly related to the well-known ℒ-class. It is shown that distributions in the ℳ-class cannot have an undesirable property recently observed in an example of an ℒ-class distribution by Klar (2002). Moreover, it is shown how these and related classes of life distributions can be characterized by expected remaining lifetimes after a family of random times, thus extending the notion of NBUE. We give examples of ℳ-class distributions by using simple sufficient conditions, and we derive reliability bounds for distributions in this class.