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 [1]. 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 [3]. See also Random variables, transformations of.

References

[1]	E.A. Cornish, R.A. Fisher, "Moments and cumulants in the specification of distributions" Rev. Inst. Internat. Statist. , 5 (1937) pp. 307–320
[2]	M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 3. Design and analysis , Griffin (1969)
[3]	L.N. Bol'shev, "Asymptotically Pearson transformations" Theor. Probab. Appl. , 8 (1963) pp. 121–146 Teor. Veroyatnost. i Primenen. , 8 : 2 (1963) pp. 129–155

Comments

For the methods of using an Edgeworth expansion to obtain (2) (see also Edgeworth series), see also [a1].

References

[a1]	P.J. Bickel, "Edgeworth expansions in non parametric statistics" Ann. Statist. , 2 (1974) pp. 1–20
[a2]	N.L. Johnson, S. Kotz, "Distributions in statistics" , 1 , Houghton Mifflin (1970)