# Difference between revisions of "Cornish-Fisher expansion"

An asymptotic expansion of the quantiles of a distribution (close to the normal standard one) in terms of the corresponding quantiles of the standard normal distribution, in powers of a small parameter. It was studied by E.A. Cornish and R.A. Fisher . If is a distribution function depending on as a parameter, if is the normal distribution function with parameters , and if as , then, subject to certain assumptions on , the Cornish–Fisher expansion of the function (where is the function inverse to ) has the form (1)

where the are certain polynomials in . Similarly, one defines the Cornish–Fisher expansion of the function ( being the function inverse to ) in powers of : (2)

where the are certain polynomials in . Formula (2) is obtained by expanding in a Taylor series about the point and using the Edgeworth expansion. Formula (1) is the inversion of (2).

If is a random variable with distribution function , then the variable is normally distributed with parameters , and, as follows from (2), approximates the distribution function of the variable as better than it approximates . If has zero expectation and unit variance, then the first terms of the expansion (1) have the form Here , , with the -th cumulant of , , , , and with the Hermite polynomials, defined by the relation Concerning expansions for random variables obeying limit laws from the family of Pearson distributions see . See also Random variables, transformations of.

How to Cite This Entry:
Cornish-Fisher expansion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cornish-Fisher_expansion&oldid=14424
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article