Peter Reinhard Hansen, Chen Tong
arXiv 31 Oct 2024 · Econometrics
arXiv:2410.23587 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Hansen, P. R. and Tong, C (2025) A unifying integral representation of the Gamma function and its reciprocal self | 0.644 | 2 | 2 | 100% |
| 2 | Meng, X.-L (2005) From unit root to Stein's estimator to Fisher's k statistics: If you have a moment, I can tell you more | 0.585 | 3 | 1 | 100% |
| 3 | Schürger, K (2002) Laplace transforms and suprema of stochastic processes | 0.511 | 3 | 2 | 33% |
| 4 | Cressie, N. and Borkent, M (1986) The moment generating function has its moments | 0.511 | 2 | 2 | 50% |
| 5 | Kawata, T (1972) Fourier analysis in probability theory | 0.511 | 2 | 2 | 50% |
| 6 | Laue, G (1980) Remarks on the relation between fractional moments and fractional derivatives of characteristic functions | 0.511 | 2 | 2 | 50% |
| 7 | Barndorff-Nielsen, O. E. and Stelzer, R (2005) Absolute moments of generalized hyperbolic distributions and approximate scaling of normal inverse gaussian lévy processes | 0.511 | 2 | 1 | 100% |
| 8 | Huang, X. and Oosterlee, C. W (2011) Saddlepoint approximations for expectations and an application to CDO pricing | 0.511 | 2 | 1 | 100% |
| 9 | Kim, Y. S., Rachev, S. T., Bianchi, M. L., and Fabozzi, F. J (2010) Computing VaR and AVaR in infinitely divisible distributions | 0.511 | 2 | 1 | 100% |
| 10 | Andersen, T. G., Bollerslev, T., Frederiksen, P., and rregaard Niels… (2010) Continuous-time models, realized volatilities, and testable distributional implications for daily stock returns | 0.405 | 1 | 1 | 100% |
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