Cornish-Fisher Asymptotic Expansion

(1)

where

(2)

where

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

and combinations of Hermite polynomials.

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.