Moments by Integrating the Moment-Generating Function

ArXiv ID: 2410.23587 “View on arXiv”

Authors: Unknown

Abstract

We introduce a novel method for obtaining a wide variety of moments of any random variable with a well-defined moment-generating function (MGF). We derive new expressions for fractional moments and fractional absolute moments, both central and non-central moments. The expressions are relatively simple integrals that involve the MGF, but do not require its derivatives. We label the new method CMGF because it uses a complex extension of the MGF and can be used to obtain complex moments. We illustrate the new method with three applications where the MGF is available in closed-form, while the corresponding densities and the derivatives of the MGF are either unavailable or very difficult to obtain.

Keywords: Moment Generating Function (MGF), Fractional Moments, Complex Moments (CMGF), Random Variables, Statistical Moments

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper presents a novel mathematical framework with advanced complex analysis and integral transforms to derive new expressions for moments, including complex and fractional moments, which is highly mathematically dense. However, the applications are purely theoretical demonstrations (e.g., normal-inverse Gaussian, Heston-Nandi GARCH, HARG) without any code, backtested performance metrics, or implementation details for empirical data, placing it in the Lab Rats quadrant.
  flowchart TD
    A["Research Goal: Develop method for moments without MGF derivatives"] --> B{"Key Methodology"}
    B --> C["Extend MGF to Complex Plane"]
    C --> D["Derive Integral Expressions<br>for Fractional & Complex Moments"]
    
    E["Data: Closed-form MGFs<br>from 3 Applications"] --> F["Computational Process"]
    
    subgraph F [" "]
        F1["Apply CMGF Formulas"]
        F2["Integrate via<br>Complex Analysis"]
    end
    
    F --> G["Key Findings/Outcomes"]
    
    G --> H["Novel CMGF Method<br>Derives moments without derivatives"]
    G --> I["New Formulas for<br>Fractional & Absolute Moments"]
    G --> J["Validated on 3 Cases<br>where density/derivatives unavailable"]