On the Study of Weighted Fractional Cumulative Residual Inaccuracy and its Dynamical Version with Applications

Abstract

In recent years, there has been a growing interest in information measures that quantify inaccuracy and uncertainty in systems. In this paper, we introduce a novel concept called the Weighted Fractional Cumulative Residual Inaccuracy (WFCRI). We develop several fundamental properties of WFCRI and establish important bounds that reveal its analytical behavior. Further, we examine the behavior of WFCRI under a mixture hazard model. A dynamic version of WFCRI also proposed and studied its behavior under proportional hazard rate model. An empirical estimation method for WFCRI under the proportional hazard rate model framework is also proposed, and its performance is evaluated through simulation studies. Finally, we demonstrate the utility of WFCRI measure in characterizing chaotic dynamics by applying it to the Ricker and cubic maps. The proposed measure is also applied to real data to assess the uncertainty.

Figures (10)

• Figure 1: Graphical illustration of the WFGCRI for a mixture of three exponential distributions with different parameter settings (Example 2.1).
• Figure 2: Plots of DWFGCRI for exponential and Rayleigh distributions for different parameter values and weight functions: (a) $\psi(w)=w$ and (b) $\psi(w)=w^2$ (Example 3.1).
• Figure 3: Graphical representations of DWFGCRI under PO model with two different weight functions: (a) $\psi(w)=w^{0.3}$ and (b) $\psi(w)=w^{2.4}$ (Example 3.3).
• Figure 4: Graphical presentation of simulation study of WFGCRI.
• Figure 5: Bifurcation diagrams of Ricker (left) and Tent maps (right).

Theorems & Definitions (25)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof