🤖 AI Summary
Conventional moment computation relies on analytic derivatives of probability density functions (PDFs) or moment-generating functions (MGFs), rendering it inapplicable to fractional, complex-order, and central/non-central moments when PDFs are unavailable or MGF derivatives are intractable. Method: This paper proposes the Complex-extended Moment Generating Function (CMGF) integration method—a novel framework that requires only integrability of the MGF over a contour in the complex plane. By leveraging analytic continuation and numerical contour integration, CMGF directly computes arbitrary real-order, complex-order, absolute, central, and non-central moments without invoking PDFs or MGF derivatives. Contribution/Results: As the first general-purpose MGF-based moment computation framework relying on integration rather than differentiation, CMGF is validated across three canonical scenarios where closed-form MGFs exist but PDFs or their derivatives do not. It significantly simplifies high-order and non-integer moment evaluation, enhancing efficiency in statistical inference and stochastic modeling.
📝 Abstract
We introduce a novel method for obtaining a wide variety of moments of a 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 new moment 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.