Some generalized information and divergence generating functions: properties, estimation, validation, and applications

Abstract We propose Rényi information generating function (RIGF) and discuss its properties. A connection between the RIGF and the diversity index is proposed for discrete-type random variables. The relation between the RIGF and Shannon entropy of order q > 0 is established and several bounds are obtained. The RIGF of escort distribution is derived. Furthermore, we introduce the Rényi divergence information generating function (RDIGF) and discuss its effect under monotone transformations. We present nonparametric and parametric estimators of the RIGF. A simulation study is carried out and a real data relating to the failure times of electronic components is analyzed. A comparison study between the nonparametric and parametric estimators is made in terms of the standard deviation, absolute bias, and mean square error. We have observed superior performance for the newly proposed estimators. Some applications of the proposed RIGF and RDIGF are provided. For three coherent systems, we calculate the values of the RIGF and other well-established uncertainty measures, and similar behavior of the RIGF is observed. Further, a study regarding the usefulness of the RDIGF and RIGF as model selection criteria is conducted. Finally, three chaotic maps are considered and then used to establish a validation of the proposed information generating function.