Dynamic generalized information measures

In many reliability and survival analysis problems the current age of an item under study must be taken into account by information measures of the lifetime distribution. Kullback-Leibler information and Shannon entropy have been considered in this context, which led to information measures that depend on time, and thus are dynamic. This paper develops dynamic information divergence and entropy of order α, also known as Rényi information and entropy, which for α=1 give the Kullback-Leibler information and Shannon entropy, respectively. We give characterizations of the proportional hazards model, the exponential distribution, and Generalized Pareto distributions in terms of dynamic Rényi information and entropy. It is also shown that dynamic Rényi entropy uniquely determines distributions that have monotone densities. A result that relates dynamic Rényi entropy and hazard rate orderings is given. This result leads to a Maximum Dynamic Entropy of order α formulation and characterizations of some well-known lifetime models. A dynamic entropy hazard rate inequality is developed as an analog of the well-known entropy moment inequality.