标题： 分数量化残差熵及其在金融数据中的应用 显示英文标题

标题： Fractional Cumulative Residual Entropy in the Quantile Framework and its Applications in the Financial Data

摘要： 分数残差熵（FCRE）是分析复杂系统的一个有力工具。大多数关于寿命随机变量的FCRE的理论结果和应用都是基于分布函数的方法。然而，在某些情况下，分布函数可能无法以显式形式获得，但具有闭合形式的分位数函数（QF），这是表示概率分布的一种替代方法。受此启发，本研究引入了基于分位数的FCRE及其动态版本，并探讨了它们在不同应用领域的各种特性和实用性。 显示英文摘要

摘要： 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.