Generalization of the Alpha-Stable Distribution with the Degree of Freedom

ArXiv ID: 2405.04693 “View on arXiv”

Authors: Unknown

Abstract

A Wright function based framework is proposed to combine and extend several distribution families. The $α$-stable distribution is generalized by adding the degree of freedom parameter. The PDF of this two-sided super distribution family subsumes those of the original $α$-stable, Student’s t distributions, as well as the exponential power distribution and the modified Bessel function of the second kind. Its CDF leads to a fractional extension of the Gauss hypergeometric function. The degree of freedom makes possible for valid variance, skewness, and kurtosis, just like Student’s t. The original $α$-stable distribution is viewed as having one degree of freedom, that explains why it lacks most of the moments. A skew-Gaussian kernel is derived from the characteristic function of the $α$-stable law, which maximally preserves the law in the new framework. To facilitate such framework, the stable count distribution is generalized as the fractional extension of the generalized gamma distribution. It provides rich subordination capabilities, one of which is the fractional $χ$ distribution that supplies the needed ‘degree of freedom’ parameter. Hence, the “new” $α$-stable distribution is a “ratio distribution” of the skew-Gaussian kernel and the fractional $χ$ distribution. Mathematically, it is a new form of higher transcendental function under the Wright function family. Last, the new univariate symmetric distribution is extended to the multivariate elliptical distribution successfully.

Keywords: stable distributions, risk modeling, heavy tails, multivariate distributions, fractional calculus

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is extremely heavy on advanced mathematical theory involving fractional calculus, Wright functions, and complex probability distributions, but lacks any implementation details, backtests, or empirical data analysis.

  flowchart TD
    A["Research Goal<br>Generalize α-stable distribution<br>by adding degree of freedom"] --> B["Methodology: Wright Function Framework"]
    B --> C["Input: Existing Distributions<br>(α-stable, Student's t, Exp. Power)"]
    C --> D["Computation: Derive Skew-Gaussian Kernel<br>via Characteristic Function"]
    D --> E["Computation: Generalize Stable Count<br>to Fractional χ Distribution"]
    E --> F["Synthesis: Ratio Distribution<br>Kernel / Fractional χ"]
    F --> G["Outcome: New 2-Sided Super Distribution<br>Valid moments & Heavy tails"]
    G --> H["Outcome: Multivariate Elliptical Extension<br>via Subordination"]