On cumulative and relative cumulative past information generating function
Santosh Kumar Chaudhary, Nitin Gupta, Achintya Roy
TL;DR
The paper develops the cumulative past information generating function (CPIG) and relative CPIG (RCPIG) as new tools for quantifying past uncertainty, linking them to generalized cumulative past entropy (GCPE) and extropy. It establishes CPIG-based stochastic orders, convolution bounds for sums of independent variables, and multiple inequalities that relate CPIG to Shannon entropy and CPE, along with empirical estimation methods. It further expands the framework through Jensen-type divergences, including JCPIG, JFCPE, and Jensen-cumulative past Taneja entropies, enabling mixture and multi-population analyses. Collectively, the work broadens the landscape of information measures for past information, offering ordering, decomposition, estimation, and divergence-based comparison methods with potential applications in reliability, economics, and statistical inference. The introduced measures provide a cohesive toolkit for comparing distributions on their past behavior and for developing further Jensen-type CPIG-based concepts.
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.