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Formulas of Absolute Moments

Author

Listed:
  • Gwo Dong Lin

    (Hwa-Kang Xing-Ye Foundation
    Academia Sinica)

  • Chin-Yuan Hu

    (National Changhua University of Education)

Abstract

The absolute moments of probability distributions are much more complicated than conventional ones. By using a direct and simpler approach, we retreat Hsu’s (J. Chinese Math. Soc. N.S. 1, 257–280, 1951) formulas in terms of the characteristic function (which have been ignored in the literature) and provide some new results as well. The case of nonnegative random variables is also investigated through both characteristic function and Laplace–Stieltjes transform. Besides, we prove that the distribution of a nonnegative random variable with a finite fractional moment can be completely determined by a proper subset of the translated fractional moments. This improves significantly Hall’s (Z. Wahrsch. Verw. Gebiete 62, 355–359, 1983) result for distributions on the right half-line.

Suggested Citation

  • Gwo Dong Lin & Chin-Yuan Hu, 2021. "Formulas of Absolute Moments," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 476-495, February.
    • Handle: RePEc:spr:sankha:v:83:y:2021:i:1:d:10.1007_s13171-019-00196-x
    DOI: 10.1007/s13171-019-00196-x
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    References listed on IDEAS

    as
    1. Ole Eiler Barndorff‐Nielsen & Robert Stelzer, 2005. "Absolute Moments of Generalized Hyperbolic Distributions and Approximate Scaling of Normal Inverse Gaussian Lévy Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 617-637, December.
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