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 $F ( x, t)$ is a distribution function depending on $t$ as a parameter, if $\Phi ( x)$ is the normal distribution function with parameters $( 0, 1)$, and if $F ( x, t) \rightarrow \Phi ( x)$ as $t \rightarrow 0$, then, subject to certain assumptions on $F ( x, t)$, the Cornish–Fisher expansion of the function $x = F ^ {-} 1 [ \Phi ( z), t]$( where $F ^ {-} 1$ is the function inverse to $F$) has the form

$$\tag{1 } x = z + \sum _ {i = 1 } ^ { {m } - 1 } S _ {i} ( z) t ^ {i} + O ( t ^ {m} ),$$

where the $S _ {i} ( z)$ are certain polynomials in $z$. Similarly, one defines the Cornish–Fisher expansion of the function $z = \Phi ^ {-} 1 [ F ( x, t)]$( $\Phi ^ {-} 1$ being the function inverse to $\Phi$) in powers of $x$:

$$\tag{2 } z = x + \sum _ {i = 1 } ^ { {m } - 1 } Q _ {i} ( x) t ^ {i} + O ( t ^ {m} ),$$

where the $Q _ {i} ( x)$ are certain polynomials in $x$. Formula (2) is obtained by expanding $\Phi ^ {-} 1$ in a Taylor series about the point $\Phi ( x)$ and using the Edgeworth expansion. Formula (1) is the inversion of (2).

If $X$ is a random variable with distribution function $F ( x, t)$, then the variable $Z = Z ( X) = \Phi ^ {-} 1 [ F ( X , t) ]$ is normally distributed with parameters $( 0, 1)$, and, as follows from (2), $\Phi ( x)$ approximates the distribution function of the variable

$$\overline{Z}\; = \ X + \sum _ {i = 1 } ^ { {m } - 1 } Q _ {i} ( X) t ^ {i}$$

as $t \rightarrow 0$ better than it approximates $F ( x, t)$. If $X$ has zero expectation and unit variance, then the first terms of the expansion (1) have the form

$$x = z + [ \gamma _ {1} h _ {1} ( z)] + [ \gamma _ {2} h _ {2} ( z) + \gamma _ {1} ^ {2} h _ {3} ( z)] + \dots .$$

Here $\gamma _ {1} = {\kappa _ {3} / \kappa _ {2} } ^ {3/2}$, $\gamma _ {2} = \kappa _ {4} / \kappa _ {2} ^ {2}$, with $\kappa _ {r}$ the $r$- th cumulant of $X$, $h _ {1} ( z) = H _ {2} ( z)/6$, $h _ {2} ( z) = H _ {3} ( z) / 24$, $h _ {3} ( z) = - [ 2H _ {3} ( z) + H _ {1} ( z)]/36$, and with $H _ {r} ( z)$ the Hermite polynomials, defined by the relation

$$\phi ( z) H _ {r} ( z) = \ (- 1) ^ {r} \frac{d ^ {r} \phi ( z) }{dz ^ {r} } \ \ ( \phi ( z) = \Phi ^ \prime ( z)).$$

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)