Computational Statistics

, Volume 17, Issue 4, pp 507–515 | Cite as

Computing Relations between Moments and Cumulants

  • Qi Zheng


It is sometimes desirable to have easily accessible formulae expressing moments in terms of cumulants and vice versa. This paper offers a Mathematica implementation of two methods for generating such formulae, a method based on Kendall’s operator and an elementary method of equating coefficients of generating functions.


symbolic computing rewrite rule Kendall’s operator cumulant moment 



This work originated from my participation in the 1995–1996 Mathematica Visiting Scholar Program sponsored by Wolfram Research Inc. I am indebted to Dr. K. Hutcheson for kindly providing me with a copy of Kratky et al. (1972). Several improvements in the presentation of the material in this article have resulted from suggestions by co-editor h. j. Newton, an associate editor, and a referee.


  1. Cook, M. B. (1951), ‘Bi-variate k-statistics and cumulants of their joint sampling distribution’, Biometrika 38, 179–195.MathSciNetCrossRefGoogle Scholar
  2. Fisher, R. A. (1928), ‘Moments and product moments of sampling distributions’, Proc. London Math. Soc. Series 2 30, 199–238.MathSciNetzbMATHGoogle Scholar
  3. Kendall, M. G. (1940), ‘The derivation of multivariate sampling formulae from univariate formulae by symbolic operation’, Annals of Eugenics 10, 392–402.MathSciNetCrossRefGoogle Scholar
  4. Kratky, J., Reinfelds, J., Hutcheson, K. & Shenton, L. R. (1972), Tables of crude moments expressed in terms of cumulants, Technical report, University of Georgia, Athens, Georgia.Google Scholar
  5. Smith, P. J., Shenton, L. R. (1995), ‘A recursive formulation of the old problem of obtaining moments from cumulants and vice versa’, The American Statistician 49, 217–218.MathSciNetGoogle Scholar
  6. Stuart, A. & Ord, J. K. (1994), Kendall’s Advanced Theory of Statistics, Volume 1: Distribution Theory, 6 edn, Arnold, London.zbMATHGoogle Scholar
  7. Zheng, Q. (1998), Computing relations between statistical moments and cumulants, ‘Proceedings of the Computational Statistics Section’, American Statistical Association, pp. 165–170.Google Scholar

Copyright information

© Physica-Verlag 2002

Authors and Affiliations

  • Qi Zheng
    • 1
  1. 1.Division of Biometry and Risk AssessmentNational Center for Toxicological ResearchJeffersonUSA

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