On cumulative entropies

In analogy with the cumulative residual entropy recently proposed by Wang, we introduce and study the cumulative entropy, which is a new measure of information alternative to the classical differential entropy. We show that the cumulative entropy of a random lifetime can be expressed as the expectation of its mean inactivity time. Hence, our measure is particularly suitable to describe the information in problems related to ageing properties of reliability theory based on the past and on the inactivity times. Our results include various upper and lower bounds to the cumulative entropy, its connection to the proportional reversed hazards model, and the study of its dynamic version that is shown to be increasing if the mean inactivity time is increasing.