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Difference between revisions of "Cornish-Fisher expansion"

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Line 18: Line 18:
 
as  $  t \rightarrow 0 $,  
 
as  $  t \rightarrow 0 $,  
 
then, subject to certain assumptions on  $  F ( x, t) $,  
 
then, subject to certain assumptions on  $  F ( x, t) $,  
the Cornish–Fisher expansion of the function  $  x = F  ^ {-} 1 [ \Phi ( z), t] $(
+
the Cornish–Fisher expansion of the function  $  x = F  ^ {-1} [ \Phi ( z), t] $
where  $  F  ^ {-} 1 $
+
(where  $  F  ^ {-1} $
 
is the function inverse to  $  F  $)  
 
is the function inverse to  $  F  $)  
 
has the form
 
has the form
  
 
$$ \tag{1 }
 
$$ \tag{1 }
x  =  z + \sum _ {i = 1 } ^ { {m - 1 }
+
x  =  z + \sum _ {i = 1 } ^ { m   - 1 }
 
S _ {i} ( z) t  ^ {i} + O ( t  ^ {m} ),
 
S _ {i} ( z) t  ^ {i} + O ( t  ^ {m} ),
 
$$
 
$$
Line 30: Line 30:
 
where the  $  S _ {i} ( z) $
 
where the  $  S _ {i} ( z) $
 
are certain polynomials in  $  z $.  
 
are certain polynomials in  $  z $.  
Similarly, one defines the Cornish–Fisher expansion of the function  $  z = \Phi  ^ {-} 1 [ F ( x, t)] $(
+
Similarly, one defines the Cornish–Fisher expansion of the function  $  z = \Phi  ^ {-1} [ F ( x, t)] $(
$  \Phi  ^ {-} 1 $
+
$  \Phi  ^ {-1} $
 
being the function inverse to  $  \Phi $)  
 
being the function inverse to  $  \Phi $)  
 
in powers of  $  x $:
 
in powers of  $  x $:
  
 
$$ \tag{2 }
 
$$ \tag{2 }
z  =  x + \sum _ {i = 1 } ^ { {m } - 1 }
+
z  =  x + \sum _ {i = 1 } ^ { m  - 1 }
 
Q _ {i} ( x) t  ^ {i} + O ( t  ^ {m} ),
 
Q _ {i} ( x) t  ^ {i} + O ( t  ^ {m} ),
 
$$
 
$$
Line 42: Line 42:
 
where the  $  Q _ {i} ( x) $
 
where the  $  Q _ {i} ( x) $
 
are certain polynomials in  $  x $.  
 
are certain polynomials in  $  x $.  
Formula (2) is obtained by expanding  $  \Phi  ^ {-} 1 $
+
Formula (2) is obtained by expanding  $  \Phi  ^ {-1} $
 
in a Taylor series about the point  $  \Phi ( x) $
 
in a Taylor series about the point  $  \Phi ( x) $
 
and using the Edgeworth expansion. Formula (1) is the inversion of (2).
 
and using the Edgeworth expansion. Formula (1) is the inversion of (2).
Line 48: Line 48:
 
If  $  X $
 
If  $  X $
 
is a random variable with distribution function  $  F ( x, t) $,  
 
is a random variable with distribution function  $  F ( x, t) $,  
then the variable  $  Z = Z ( X) = \Phi  ^ {-} 1 [ F ( X , t) ] $
+
then the variable  $  Z = Z ( X) = \Phi  ^ {-1} [ F ( X , t) ] $
 
is normally distributed with parameters  $  ( 0, 1) $,  
 
is normally distributed with parameters  $  ( 0, 1) $,  
 
and, as follows from (2),  $  \Phi ( x) $
 
and, as follows from (2),  $  \Phi ( x) $
Line 55: Line 55:
 
$$  
 
$$  
 
\overline{Z}\;  = \  
 
\overline{Z}\;  = \  
X + \sum _ {i = 1 } ^ { {m } - 1 }
+
X + \sum _ {i = 1 } ^ { m  - 1 }
 
Q _ {i} ( X) t  ^ {i}
 
Q _ {i} ( X) t  ^ {i}
 
$$
 
$$

Latest revision as of 16:41, 15 January 2021


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)
How to Cite This Entry:
Cornish-Fisher expansion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cornish-Fisher_expansion&oldid=46519
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article