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Moments by Integrating the Moment-Generating Function

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  • Peter Reinhard Hansen
  • Chen Tong

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.

Suggested Citation

  • Peter Reinhard Hansen & Chen Tong, 2024. "Moments by Integrating the Moment-Generating Function," Papers 2410.23587, arXiv.org, revised Oct 2025.
    • Handle: RePEc:arx:papers:2410.23587
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    References listed on IDEAS

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