A Note on Some Functional Relationships Involving the Mean Inactivity Time Order

Stochastic comparisons from inactivity times have been used in several reliability problems. The inactivity time represents the interval time elapsed after an event occurs until the time of its observation, and it is useful to predict the exact times of occurrence of events. We obtain a characterization of the reversed hazard rate order, and a functional relationship involving the mean inactivity time order for non-negative random variables that model lifetimes. Also, we obtain comparison results of two random sums based on their mean inactivity times, which allow us to select a better engineering system subjected to a general shock model. Our results provide other applications in reliability, statistics, and risk theory.