Weighted Cumulative Information Generating Functions

Abstract The weighted cumulative information generating function (wcigf) is introduced and interrelations with various weighted and non-weighted measures of uncertainty are explored in detail. It turns out that the wcigf represents the generating function of several weighted cumulative versions of the Shannon entropy. Additionally, several properties and representations of wcigf are shown, and connections with order statistics are studied. Further applications arise by defining weighted efficiency Gini-based indices for systems in terms of ratios of wcigf’s.