Moments by the Integrating the Moment-Generating Function

Peter Hansen and Chen Tong

Papers from arXiv.org

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.

Date: 2024-10, Revised 2025-03
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