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Cornish-Fisher Asymptotic Expansion

A Cornish-Fisher asymptotic expansion is given by

y approx m+sigmaw,
(1)

where

w=x+[gamma_1h_1(x)]+[gamma_2h_2(x)+gamma_1^2h_(11)(x)]+[gamma_3h_3(x)+gamma_1gamma_2h_(12)(x)+gamma_1^3h_(111)(x)]+[gamma_4h_4(x)+gamma_2^2h_(22)(x)+gamma_1gamma_3h_(13)(x)+gamma_1^2gamma_2h_(112)(x)+gamma_1^4h_(1111)(x)]+...,
(2)

where

h_1(x)=1/6He_2(x)
(3)
h_2(x)=1/(24)He_3(x)
(4)
h_(11)(x)=-1/(36)[2He_3(x)+He_1(x)]
(5)
h_3(x)=1/(120)He_4(x)
(6)
h_(12)(x)=-1/(24)[He_4(x)+He_2(x)]
(7)
h_(111)(x)=1/(324)[12He_4(x)+19He_2(x)]
(8)
h_4(x)=1/(720)He_5(x)
(9)
h_(22)(x)=-1/(384)[3He_5(x)+6He_3(x)+2He_1(x)]
(10)
h_(13)(x)=-1/(180)[2He_5+3He_3(x)]
(11)
h_(112)(x)=1/(288)[14He_5(x)+37He_3(x)+8He_1(x)]
(12)
h_(1111)(x)=-1/(7776)[252He_5(x)+832He_3(x)+227He_1(x)]
(13)

and combinations of Hermite polynomials.

See also

Charlier Series, Edgeworth Series

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 935, 1972.Cornish, E. A. and Fisher, R. A. "Moments and Cumulants in the Specification of Distributions." Extrait de la Revue de l'Institute International de Statistique 4, 1-14, 1937. Reprinted in Fisher, R. A. Contributions to Mathematical Statistics. New York: Wiley, 1950.Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.Wasow, W. "On the Asymptotic Transformation of Certain Distributions into the Normal Distribution." Proceedings of Symposia in Applied Mathematica VI, Numerical Analysis. New York: McGraw-Hill, pp. 251-259, 1956.

Referenced on Wolfram|Alpha

Cornish-Fisher Asymptotic Expansion

Cite this as:

Weisstein, Eric W. "Cornish-Fisher Asymptotic Expansion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Cornish-FisherAsymptoticExpansion.html

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