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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) |
Cornish-Fisher expansion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cornish-Fisher_expansion&oldid=14424