Weighted fractional generalized cumulative past entropy and its properties

Abstract: In this paper, we introduce weighted fractional generalized cumulative past entropy of a nonnegative absolutely continuous random variable with bounded support. Various properties of the proposed weighted fractional measure are studied. Bounds and stochastic orderings are derived. A connection between the proposed measure and the left-sided Riemann-Liouville fractional integral is established. Further, the proposed measure is studied for the proportional reversed hazard rate models. Next, a nonparametric estimator of the weighted fractional generalized cumulative past entropy is suggested based on empirical distribution function. Various examples with a real life data set are considered for the illustration purposes. Finally, large sample properties of the proposed empirical estimator are studied.

• The paper introduces WFGCPE, a novel entropy measure that incorporates weighted functions and fractional calculus for enhanced statistical analysis.
• It establishes theoretical properties including bounds, stochastic ordering, and a central limit theorem for large-sample evaluations.
• Empirical examples from medical and industrial datasets demonstrate WFGCPE's practical ability to capture both quantitative and qualitative system variations.

Weighted Fractional Generalized Cumulative Past Entropy

Introduction and Motivation

The concept of entropy has been a cornerstone in statistical mechanics and information theory, serving as a measure of uncertainty or unpredictability in a system. Traditional entropy measures, such as those introduced by Boltzmann, Gibbs, and Shannon, have seen various generalizations to address different analytical requirements. This paper introduces a novel formulation, the Weighted Fractional Generalized Cumulative Past Entropy (WFGCPE), building upon previous definitions of fractional and weighted entropies.

The motivation behind this introduction is twofold: capturing more nuanced statistical properties of systems by incorporating fractional derivatives and addressing the context-dependent nature of certain statistical analyses through weighted measures. In particular, WFGCPE is designed to handle non-negative continuous random variables with bounded support, providing insights that account for both probabilistic occurrence and qualitative factors of events, thereby extending the utility of entropy measures in statistical analysis.

Properties and Theoretical Insights

Definition and Basic Properties: The WFGCPE for a nonnegative random variable $X$ with bounded support is defined under a general non-negative weight function $\psi(x)$, adding a layer of contextual adaptability through weight application. It extends previous entropy measures by using left-sided Riemann-Liouville fractional integrals, offering a bridge to concepts in fractional calculus. Essentially, this generalization captures both the "memory" aspect of prior distribution and fractional order dependencies.

Stochastic Ordering Implications: The paper explores how the WFGCPE behaves under various conditions of stochastic ordering. The measure respects the dispersive order, highlighting its potential to differentiate distributions not discernable with traditional measures. Some results indicate shift-dependence, where changes in variable location or scale influence the entropy, aligning with weighted and fractional modifications.

Bounds and Limit Theorems: Mathematical results establish key bounds for WFGCPE, showing the dual behavorial boundaries for increasing and decreasing functions $\psi$. This contextual dependency echoes through the bounded relationships for stochastic models, such as those seen in proportional hazards contexts. Furthermore, as $n \rightarrow \infty$, empirical parameters of the proposed measure follow a central limit theorem, thereby reinforcing the measure’s statistical robustness in large sample evaluations.

Practical Implications and Examples

The proposed measure's main utilities lie in practical situations where understanding system variability in response to changes in qualitative attributes matter. The empirical results based on real datasets, such as lifetimes of medical patients or engineered systems, showcase the significant insights offered by this approach.

Detailed examples illustrate calculating means and variances of WFGCPE, providing crucial information for decision-making processes in clinical, industrial, and experimental studies. These examples validate the WFGCPE as a tool for quantitative assessment augmented with qualitative contextuality, often essential for thorough statistical analysis in complex systems.

Conclusion

The WFGCPE introduced in this study extends the analytical versatility of entropy measures by combining fractional calculus and weighted statistical frameworks. Its application across different stochastic orders and its adaptability to varying qualitative inputs make it a powerful tool for researchers dealing with complex systems where both data-driven and theoretical insights are critical. Future work can expand on empirical studies and refine estimation techniques, particularly in multivariate setups and higher-dimensional data settings, aiming to enhance entropy-related analysis further in machine learning and AI applications.