Asymmetric $q$-Gaussian Distributions from Tsallis Maximum Entropy

Paper: arXiv 2610.16663

Authors: Francesco Cesarone, Massimiliano Corradini, Nicolò Giunta, Serena Rocci

Abstract

Within the framework of non-extensive statistical mechanics, $q$-Gaussian distributions arise as maximum-entropy solutions of Tsallis entropy under suitable escort moment constraints. Although widely used in physics, biology, geology, finance, and related fields, standard $q$-Gaussians are intrinsically symmetric and cannot capture asymmetric fluctuations often observed in empirical data. We ask whether asymmetry can arise from the Tsallis maximum-entropy principle without externally skewing or joining distributions. We show that this is achieved by retaining the Tsallis entropy and imposing a global escort mean together with distinct downside and upside escort semi-variances. The key step is to reformulate the problem of maximum Tsallis entropy in terms of the escort density. The nonlinear escort constraints then become affine, while maximizing Tsallis entropy becomes equivalent to maximizing a strictly concave functional over a convex feasible set. Consequently, any admissible stationary solution is the unique global maximizer of the corresponding variational problem. The finite-sample approximation properties of the proposed asymmetric $q$-Gaussian are studied through Monte Carlo experiments using synthetic data generated from Azzalini skew-Student distributions. Compared with the classical symmetric $q$-Gaussian, the asymmetric specification systematically provides a closer approximation to synthetic asymmetric heavy-tailed distributions. Finally, we fit the model to return series from major equity indices, a commodity, and cryptocurrencies, and compare its in-sample goodness of fit with that of a symmetric $q$-Gaussian. The results show that the asymmetric model offers a closer in-sample description of unconditional return distributions, most clearly in the tails.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a novel mathematical framework for asymmetric q-Gaussian distributions derived from Tsallis maximum entropy, addressing a significant limitation of previous models. It combines rigorous mathematical derivations with empirical validation on both synthetic and real financial data, demonstrating its practical applicability. The methodology is clearly articulated, though full reproducibility would benefit from explicit code/data sharing.

Research Flowchart

  flowchart TD
    A[Research Goal: Generate Asymmetric q-Gaussian Distributions from Tsallis Max Entropy] --> B{Key Methodology: Reformulate Tsallis Max Entropy for Escort Density};
    B --> C{Constraints: Global Escort Mean + Distinct Downside & Upside Escort Semi-Variances};
    C --> D[Computational Process: Maximize Strictly Concave Functional over Convex Set (Unique Global Maximizer)];
    D --> E[Data/Inputs: Synthetic Azzalini Skew-Student Distributions & Real-World Financial Return Series];
    E --> F[Key Findings: Asymmetric q-Gaussian Systematically Approximates Asymmetric Heavy-Tailed Distributions Better];
    F --> G[Outcomes: Improved In-Sample Fit for Equity Indices, Commodity, Cryptocurrencies, Especially in Tails];

Cited References (222)

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3. Transition from lognormal to $\chi$2-superstatistics for financial time series (2016)
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6. Possible generalization of Boltzmann-Gibbs statistics (1988)
7. Introduction to Nonextensive Statistical Mechanics : Approaching a Complex World (2009) — link
8. Introduction to methods for nonlinear optimization (2023)

0 of 222 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.