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Unbiased Estimates for Products of Moments and Cumulants for Finite Populations

Author

Listed:
  • Christopher S. Withers

    (Callaghan Innovation, Lower Hutt 5011, New Zealand)

  • Saralees Nadarajah

    (Department of Mathematics, University of Manchester, Manchester M13 9PL, UK)

Abstract

Let F N be the distribution function of a finite real population of size N . Let F n be the empirical distribution function of a sample of size n drawn from the population without replacement. Let T F N be any product of the moments or cumulants of F N , let T F n denote the sample version, and let T n , N F N denote the expected value of T F n with respect to F N . We prove the following remarkable inversion principle that the expected value of T N , n F n is equal to T F N . We also obtain an explicit expression for T n , N F N for all T F N of orders up to six.

Suggested Citation

  • Christopher S. Withers & Saralees Nadarajah, 2023. "Unbiased Estimates for Products of Moments and Cumulants for Finite Populations," Mathematics, MDPI, vol. 11(17), pages 1-36, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:17:p:3720-:d:1228302
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    References listed on IDEAS

    as
    1. Eric Jondeau & Michael Rockinger, 2006. "Optimal Portfolio Allocation under Higher Moments," European Financial Management, European Financial Management Association, vol. 12(1), pages 29-55, January.
    2. Kraus, Alan & Litzenberger, Robert H, 1976. "Skewness Preference and the Valuation of Risk Assets," Journal of Finance, American Finance Association, vol. 31(4), pages 1085-1100, September.
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