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)