View source for 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 [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265001.png" /> is a distribution function depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265002.png" /> as a parameter, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265003.png" /> is the normal distribution function with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265004.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265005.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265006.png" />, then, subject to certain assumptions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265007.png" />, the Cornish–Fisher expansion of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265008.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c0265009.png" /> is the function inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650010.png" />) has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650012.png" /> are certain polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650013.png" />. Similarly, one defines the Cornish–Fisher expansion of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650015.png" /> being the function inverse to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650016.png" />) in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650017.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650019.png" /> are certain polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650020.png" />. Formula (2) is obtained by expanding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650021.png" /> in a Taylor series about the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650022.png" /> and using the Edgeworth expansion. Formula (1) is the inversion of (2).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650023.png" /> is a random variable with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650024.png" />, then the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650025.png" /> is normally distributed with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650026.png" />, and, as follows from (2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650027.png" /> approximates the distribution function of the variable

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650028.png" /></td> </tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650029.png" /> better than it approximates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650031.png" /> has zero expectation and unit variance, then the first terms of the expansion (1) have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650032.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650034.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650035.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650036.png" />-th cumulant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650040.png" />, and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650041.png" /> the Hermite polynomials, defined by the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026500/c02650042.png" /></td> </tr></table>

Concerning expansions for random variables obeying limit laws from the family of Pearson distributions see [[#References|[3]]]. See also [[Random variables, transformations of|Random variables, transformations of]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.A. Cornish,   R.A. Fisher,   "Moments and cumulants in the specification of distributions"  ''Rev. Inst. Internat. Statist.'' , '''5'''  (1937)  pp. 307–320</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,   A. Stuart,   "The advanced theory of statistics. Distribution theory" , '''3. Design and analysis''' , Griffin  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Bickel,   "Edgeworth expansions in non parametric statistics"  ''Ann. Statist.'' , '''2'''  (1974)  pp. 1–20</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.L. Johnson,   S. Kotz,   "Distributions in statistics" , '''1''' , Houghton Mifflin  (1970)</TD></TR></table>