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A new method for fast computing unbiased estimators of cumulants

  • E. Di Nardo
  • G. Guarino
  • D. SenatoEmail author
Article

Abstract

We propose new algorithms for generating k-statistics, multivariate k-statistics, polykays and multivariate polykays. The resulting computational times are very fast compared with procedures existing in the literature. Such speeding up is obtained by means of a symbolic method arising from the classical umbral calculus. The classical umbral calculus is a light syntax that involves only elementary rules to managing sequences of numbers or polynomials. The cornerstone of the procedures here introduced is the connection between cumulants of a random variable and a suitable compound Poisson random variable. Such a connection holds also for multivariate random variables.

Keywords

Univariate and multivariate k-statistics Univariate and multivariate polykays Umbral calculus 

References

  1. Andrews, D.F., Stafford, J.E.: Symbolic Computation for Statistical Inference. Oxford Statistical Science Series, vol. 21. Oxford University Press, London (2000) zbMATHGoogle Scholar
  2. Di Nardo, E., Senato, D.: Umbral nature of the Poisson random variables. In: Crapo, H., Senato, D. (eds.) Algebraic Combinatorics and Computer Science, pp. 245–266. Springer, Berlin (2001) Google Scholar
  3. Di Nardo, E., Senato, D.: An umbral setting for cumulants and factorial moments. Eur. J. Comb. 27, 394–413 (2006a) zbMATHCrossRefGoogle Scholar
  4. Di Nardo, E., Senato, D.: A symbolic method for k-statistics. Appl. Math. Lett. 19, 968–975 (2006b) zbMATHCrossRefMathSciNetGoogle Scholar
  5. Di Nardo, E., Guarino, G., Senato, D.: A unifying framework for k-statistics, polykays and their multivariate generalizations. Bernoulli 14(2), 440–468 (2008a) CrossRefzbMATHGoogle Scholar
  6. Di Nardo, E., Guarino, G., Senato, D.: Maple algorithms for polykays and multivariate polykays. Adv. Appl. Stat. 8, 19–36 (2008b) zbMATHGoogle Scholar
  7. Dressel, P.L.: Statistical seminvariants and their estimates with particular emphasis on their relation to algebraic invariants. Ann. Math. Stat. 11, 33–57 (1940) zbMATHCrossRefMathSciNetGoogle Scholar
  8. Fisher, R.A.: Moments and product moments of sampling distributions. Proc. Lond. Math. Soc. 30(2), 199–238 (1929) Google Scholar
  9. Ferreira, P.G., Magueijo, J., Silk, J.: Cumulants as non-Gaussian qualifiers. Phys. Rev. D 56, 4592–4603 (1997) CrossRefGoogle Scholar
  10. Gessel, I.: Applications of the classical umbral calculus. Algebra Univers. 49, 397–434 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  11. Grossman, R. (ed.): Symbolic Computation: Applications to Scientific Computing. Frontiers in Applied Mathematics, vol. 5. SIAM, Philadelphia (1989) zbMATHGoogle Scholar
  12. McCullagh, P.: Tensor Methods in Statistics. Monographs on Statistics and Applied Probability. Chapman & Hall, London (1987) zbMATHGoogle Scholar
  13. Rose, C., Smith, M.D.: Mathematical Statistics with Mathematica. Springer, New York (2002) zbMATHGoogle Scholar
  14. Rota, G.C., Taylor, B.D.: The classical umbral calculus. SIAM J. Math. Anal. 25, 694–711 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  15. Speed, T.P.: Cumulants and partition lattices. Aust. J. Stat. 25, 378–388 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  16. Speed, T.P.: Cumulants and partition lattices. II: Generalised k-statistics. J. Aust. Math. Soc. Ser. A 40, 34–53 (1986) zbMATHMathSciNetCrossRefGoogle Scholar
  17. Stuart, A., Ord, J.K.: Kendall’s Advanced Theory of Statistics, vol. 1. Charles Griffin and Company Limited, London (1987) zbMATHGoogle Scholar
  18. Tukey, J.W.: Some sampling simplified. J. Am. Stat. Assoc. 45, 501–519 (1950) zbMATHCrossRefMathSciNetGoogle Scholar
  19. Wang, D., Zheng, Z. (eds.): Differential Equations with Symbolic Computations. Trends in Mathematics. Birkhauser, Basel (2005) Google Scholar
  20. Zeilberger, D.: Symbolic moment calculus I.: foundations and permutation pattern statistics. Ann. Comb. 8, 369–378 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  21. Zeilberger, D.: The umbral transfer-matrix method. I: Foundations. J. Comb. Theory Ser. A 91(1–2), 451–463 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  22. Zeilberger, D.: The umbral transfer-matrix method. III: Counting animals. NY J. Math. 7, 223–231 (2001a) zbMATHMathSciNetGoogle Scholar
  23. Zeilberger, D.: The umbral transfer-matrix method. IV: Counting self-avoiding polygons and walks. Electron. J. Comb. 8(1), 17 (2001b). Research paper R28 MathSciNetGoogle Scholar
  24. Zeilberger, D.: The umbral transfer-matrix method. V: The Goulden-Jackson cluster method for infinitely many mistakes. Integers 2, 12 (2002). Paper A05 MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi della BasilicataPotenzaItaly
  2. 2.Medical SchoolUniversità Cattolica del Sacro Cuore (Rome branch)RomaItaly

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