Failure extropy, dynamic failure extropy and their weighted versions

Extropy was introduced as a dual complement of the Shannon entropy. In this investigation, we consider failure extropy and its dynamic version. Various basic properties of these measures are presented. It is shown that the dynamic failure extropy characterizes the distribution function uniquely. We also consider weighted versions of these measures. Several virtues of the weighted measures are explored. Finally, nonparametric estimators are introduced based on the empirical distribution function.