Pseudo-Distributions: Thermodynamic Geometry and an Empirical Application
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.