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

TL;DR

This paper introduces two new uncertainty measures, weighted past varentropy (WPVE) and weighted paired dynamic varentropy (WPDVE), to quantify information dispersion in past lifetimes and joint past–residual uncertainty. It develops theoretical properties, including monotone transformation behavior and upper/lower bounds via weighted Shannon entropy and the cumulative reversed hazard rate, and extends these concepts to the Proportional Reversed Hazard Rate (PRHR) model. The authors propose both kernel-based non-parametric and maximum-likelihood parametric estimators for WPVE and WPDVE, validate them through simulations, and apply them to real data (average wind speeds) with bootstrap comparisons. The work also demonstrates applications in reliability engineering, including coherent-system analyses and distortion-based bounds, highlighting the practical impact of WPVE and WPDVE for assessing uncertainty in aging and failure processes.

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.

Figures (7)

• Figure 1: Graphs for the WPVE of $(a)$ uniform distribution in Example \ref{['ex2.1']}$(i)$, $(b)$ Pareto-I distribution in Example \ref{['ex2.1']}$(ii)$, and $(c)$ exponential distribution in Example \ref{['ex2.1']}$(iii)$.
• Figure 2: Graphical plots of $e^{-(\alpha x+\beta)}$ (blue colour) and $g(x)=\frac{\gamma}{\delta}(1+x/\delta)^{-(\gamma+1)}$ (red colour) for $\alpha=2, ~\beta=1,~ \delta=1,$ and $\gamma=3$. To capture the full support $x\in(0,\infty)$, we take $x=-\log y$, where $y\in(0,1).$
• Figure 3: Plots of the WPVE for the Weibull distribution in Example \ref{['ex2.3']}$(a)$ with respect to $t$ (for fixed $\lambda$) and $(b)$ with respect to $\lambda$ (for fixed $t$).
• Figure 4: Graphs of the WPVE in Example \ref{['ex2.4']}$(a)$ with respect to $t$ (for fixed $b$) and $(b)$ with respect to $\beta$ (for fixed $t$).
• Figure 5: Plots for the WPVE $(a)$ with respect to $t$ (for fixed $a,\alpha,\beta$) and $(b)$ with respect to $a$ (for fixed $t,\alpha,\beta$) in Example \ref{['ex3.1']}.

Theorems & Definitions (37)

• Definition 2.1
• Remark 2.1
• Example 2.1
• Theorem 2.1
• proof
• Example 2.2
• Theorem 2.2
• proof
• Corollary 2.1
• Theorem 2.3