Extended gamma processes have been seen as a flexible extension of standard gamma processes in the recent reliability literature, for the purpose of cumulative deterioration modeling. The probabilistic properties of the standard gamma process have been well explored since the 1970s, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and ageing properties for the corresponding level-crossing times are now well understood. The aim of this paper is to explore similar properties for extended gamma processes and see which ones can be broadened to this new context. As a by-product, new stochastic comparisons for convolutions of gamma random variables are also obtained.

The comparison of lifetimes has been treated extensively during the last decade. A wide variety of mathematical objects have been defined, which, in reliability theory, are used to quantify ageing properties. In this work, using the equilibrium variable, we give a new viewpoint on ageing properties. Moreover, we give new bounds on the moments of series systems.

Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.