On cumulative and relative cumulative past information generating function

Abstract: In this paper, we introduce the cumulative past information generating function (CPIG) and relative cumulative past information generating function (RCPIG). We study its properties. We establish its relation with generalized cumulative past entropy (GCPE). We defined CPIG stochastic order and its relation with dispersive order. We provide the results for the CPIG measure of the convoluted random variables in terms of the measures of its components. We found some inequality relating to Shannon entropy, CPIG and GCPE. Some characterization and estimation results are also discussed regarding CPIG. We defined divergence measures between two random variables, Jensen-cumulative past information generating function(JCPIG), Jensen fractional cumulative past entropy measure, cumulative past Taneja entropy, and Jensen cumulative past Taneja entropy information measure.