伪分布：热力学几何与实证应用
Pseudo-Distributions: Thermodynamic Geometry and an Empirical Application
AI总结：
本研究基于$g$-演算构建伪分布框架，推导其热力学几何性质，并将其应用于原油价格波动分析，发现该伪分布能更好描述高频金融数据的重尾特征。
AI中文摘要：
我们开发了一种基于$g$-演算的一致伪分析框架，用于构造变形统计分布。通过单调生成函数映射标准代数运算，我们系统推导了相关的伪对数与伪指数结构。将该形式体系应用于概率密度函数和累积分布函数层面，我们引入了一类新的伪分布族，该分布族推广了非广延统计力学，可在极限情形下还原经典统计与标准非广延统计。我们利用平衡流形上的Ruppeiner度量研究了所提模型的热力学几何。围绕经典极限的微扰分析表明，在一阶近似下，热力学标量曲率仅由生成函数的形变参数决定，而非广延参数则保持解耦。为评估该框架的实证稳健性，我们将该模型用于分析西德克萨斯中质原油价格相对于其百日移动平均的日度绝对偏差。基于信息准则的模型比较显示，与标准基准相比，所提伪分布能更好地描述这些高频金融波动及其重尾特征。这些结果表明，$g$-演算为生成变形统计量及分析其几何性质提供了一种灵活且具有物理基础的数学工具。
英文摘要：
We develop a consistent pseudo-analytic framework based on $g$-calculus for constructing deformed statistical distributions. By mapping standard algebraic operations through a monotone generator function, we systematically derive the associated pseudo-logarithmic and pseudo-exponential structures. Applying this formalism, we introduce a new family of pseudo-distributions that generalizes nonextensive statistical mechanics at both the probability density and cumulative distribution levels, recovering classical and standard nonextensive statistics as limiting cases. We investigate the thermodynamic geometry of the proposed models using the Ruppeiner metric on the equilibrium manifold. A perturbative analysis around the classical limit reveals that, to leading order, the thermodynamic scalar curvature is governed solely by the generator deformation parameter, while the nonextensivity parameter remains decoupled. To evaluate the empirical robustness of the framework, we apply the model to analyze the absolute deviations of daily West Texas Intermediate crude oil prices from their hundred-day moving average. Model comparison based on information criteria demonstrates that the proposed pseudo-distributions provide a superior description of these high-frequency financial fluctuations and their heavy-tailed characteristics compared to standard benchmarks. These results suggest that $g$-calculus offers a flexible and physically grounded mathematical tool for generating deformed statistics and analyzing their geometric properties.
