Fractional Cumulative Residual Entropy in the Quantile Framework and its Applications in the Financial Data
Iona Ann Sebastian, S. M. Sunoj
TL;DR
The paper introduces the quantile-based fractional cumulative residual entropy (Q-FCRE) and its dynamic variant (Q-DFCRE) for distributions with explicit quantile functions but intractable CDFs. It defines $\mathcal{E}_{\alpha}^{Q}(X)=\int_{0}^{1} (1-p)(-\log(1-p))^{\alpha} q(p)\,dp$ and develops a nonparametric plug-in estimator, along with theoretical properties, orderings, and transformation results. The authors validate the estimator via simulations under power-Pareto and Govindarajulu quantile models and demonstrate the method's discriminative power on the logistic map's chaotic vs. periodic regimes. They further apply Q-FCRE to DJIA price returns, showing enhanced sensitivity for small $\alpha$ and highlighting its potential for monitoring financial instability. Overall, Q-FCRE provides a versatile, quantile-based uncertainty measure suitable for complex, nonstationary systems and practical financial analysis.
Abstract
Fractional cumulative residual entropy (FCRE) is a powerful tool for the analysis of complex systems. Most of the theoretical results and applications related to the FCRE of the lifetime random variable are based on the distribution function approach. However, there are situations in which the distribution function may not be available in explicit form but has a closed-form quantile function (QF), an alternative method of representing a probability distribution. Motivated by this, in the present study we introduce a quantile-based FCRE, its dynamic version and their various properties and examine their usefulness in different applied fields.