Charlier and edgeworth expansions for distributions and densities in terms of bell polynomials

• Autorzy: Withers C. S. , Nadarajah S.
• Języki publikacji: EN

Abstrakty

EN

We show that the coefficients of the Charlier differential series for distributions and densities are simply Bell polynomials in the cumulants. The same is true for the Edgeworth expansions of distributions and densities of sample means. We use this to obtain higher order extensions of these well-known series.

Słowa kluczowe

EN

Bell polynomials; Charlier series; Cornish and Fisher

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2009
• Tom: Vol. 29, Fasc. 2
• Strony: 271--280
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Withers C. S.

• Applied Mathematics Group, Industrial Research Limited Lower Hutt, New Zealand

autor

Nadarajah S.

• School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom

Bibliografia

• [1] R. N. Bhattacharya and R. R. Rao, Normal Approximation and Asymptotic Expansions, Wiley, New York 1976.
• [2] C. V. L. Charlier, Applications de la théorie des probabilités à l’astronomie, Tome II. Les applications de la théorie des probabilités aux sciences mathématiques et aux science physique, E. Borel (Ed.), Gauthier-Villars, Paris 1931.
• [3] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht 1974.
• [4] E. A. Cornish and R. A. Fisher, Moments and cumulants in the specification of distributions, Review of the International Statistical Institute 5 (1937), pp. 307-320. Reproduced in the collected papers of R. A. Fisher, volume 4.
• [5] R. A. Fisher and E. A. Cornish, The percentile points of distributions having known cumulants, Technometrics 2 (1960), pp. 209-225.
• [6] G. W. Hill and A. W. Davis, Generalised asymptotic expansions of Cornish-Fisher type, Ann. Math. Statist. 39 (1968), pp. 1264-1273.
• [7] A. Stuart and K. Ord, Kendall’s Advanced Theory of Statistics, Vol. 1, 5th edition, Griffin, London 1987.
• [8] C. S. Withers, Asymptotic expansions for distributions and quantiles with power series cumulants, J. Roy. Statist. Soc. B 46 (1984), pp. 389-396. Correction: 48 (1986), p. 258.
• [9] C. S. Withers, A simple expression for the multivariate Hermite polynomials, Statist. Probab. Lett. 47 (2000), pp. 165-169.
• [10] C. S. Withers and S. Nadarajah, Expansions for log densities of asymptotically normal estimates, Statist. Papers (2008), doi: 10.1007/s00362-008-0135-2.
• [11] C. S. Withers and S. Nadarajah, Tilted Edgeworth expansions for asymptotically normal vectors, Ann. Inst. Statist. Math. (2008), doi: 10.1007/s10463-008-0206-0.