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Charlier Series


A class of formal series expansions in derivatives of a distribution Psi(t) which may (but need not) be the normal distribution function

 Phi(t)=1/(sqrt(2pi))e^(-t^2/2)
(1)

and moments or other measured parameters. Edgeworth series are known as the Charlier series or Gram-Charlier series. Let psi(t) be the characteristic function of the function Psi(t), and gamma_r its cumulants. Similarly, let F(t) be the distribution to be approximated, f(t) its characteristic function, and kappa_r its cumulants. By definition, these quantities are connected by the formal series

 f(t)=exp[sum_(r=1)^infty(kappa_r-gamma_r)((it)^r)/(r!)]psi(t)
(2)

(Wallace 1958). Integrating by parts gives (it)^rpsi(t) as the characteristic function of (-1)^rPsi^((r))(x), so the formal identity corresponds pairwise to the identity

 F(x)=exp[sum_(r=1)^infty(kappa_r-gamma_r)((-D)^r)/(r!)]Psi(x),
(3)

where D is the differential operator. The most important case Psi(t)=Phi(t) was considered by Chebyshev (1890), Charlier (1905-06), and Edgeworth (1905).

Expanding and collecting terms according to the order of the derivatives gives the so-called Gram-Charlier A-Series, which is identical to the formal expansion of F-Psi in Hermite polynomials. The A-series converges for functions F whose tails approach zero faster than Psi^('1/2) (Cramér 1925, Wallace 1958, Szegö 1975).


See also

Cornish-Fisher Asymptotic Expansion, Edgeworth Series

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References

Charlier, C. V. L. "Über das Fehlergesetz." Ark. Math. Astr. och Phys. 2, No. 8, 1-9, 1905-06.Chebyshev, P. L. "Sur deux théorèmes relatifs aux probabilités." Acta Math. 14, 305-315, 1890.Cramér, H. "On Some Classes of Series Used in Mathematical Statistics." Proceedings of the Sixth Scandinavian Congress of Mathematicians, Copenhagen. pp. 399-425, 1925.Edgeworth, F. Y. "The Law of Error." Cambridge Philos. Soc. 20, 36-66 and 113-141, 1905.Gram, J. P. "Über die Entwicklung reeler Funktionen in Reihen mittelst der Methode der kleinsten Quadrate." J. reine angew. Math. 94, 41-73, 1883.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.

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Charlier Series

Cite this as:

Weisstein, Eric W. "Charlier Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CharlierSeries.html

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