Weighted past and paired dynamic varentropy measures, their properties and usefulness

Abstract

We introduce two uncertainty measures, say weighted past varentropy (WPVE) and weighted paired dynamic varentropy (WPDVE). Several properties of these proposed measures, including their effect under the monotone transformations are studied. An upper bound of the WPVE using the weighted past Shannon entropy and a lower bound of the WPVE are obtained. Further, the WPVE is studied for the proportional reversed hazard rate (PRHR) models. Upper and lower bounds of the WPDVE are derived. In addition, the non-parametric kernel estimates of the WPVE and WPDVE are proposed. Furthermore, the maximum likelihood estimation technique is employed to estimate WPVE and WPDVE for an exponential population. A numerical simulation is provided to observe the behaviour of the proposed estimates. A real data set is analysed, and then the estimated values of WPVE are obtained. Based on the bootstrap samples generated from the real data set, the performance of the non-parametric and parametric estimators of the WPVE and WPDVE is compared in terms of the absolute bias and mean squared error (MSE). Finally, we have reported an application of WPVE.