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m (fixing superscripts)
 
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$  f ^ { ( k) } $
 
$  f ^ { ( k) } $
is the  $  k $-
+
is the  $  k $-th derivative of  $  f $,  
th derivative of  $  f $,  
 
 
which can be represented as
 
which can be represented as
  
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\left [  
 
\left [  
\frac{x  ^ {[} 2] }{2 }
+
\frac{x  ^ {[2]} }{2 }
 
  -
 
  -
\lambda x  ^ {[} 1] +
+
\lambda x  ^ {[1]} +
  
 
\frac{\lambda  ^ {2} }{2 }
 
\frac{\lambda  ^ {2} }{2 }
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\frac{\mu _ {3} - 3 \mu _ {2} + 2 \lambda }{\lambda  ^ {3} }
 
\frac{\mu _ {3} - 3 \mu _ {2} + 2 \lambda }{\lambda  ^ {3} }
 
  \left [  
 
  \left [  
\frac{x  ^ {[} 3] }{6 }
+
\frac{x  ^ {[3]} }{6 }
 
  -
 
  -
 
{
 
{
 
\frac \lambda {2}
 
\frac \lambda {2}
  } x  ^ {[} 2] +  
+
  } x  ^ {[2]} +  
 
\frac{\lambda  ^ {2} }{2 }
 
\frac{\lambda  ^ {2} }{2 }
  x  ^ {[} 1] -  
+
  x  ^ {[1]} -  
 
\frac{\lambda  ^ {3} }{6 }
 
\frac{\lambda  ^ {3} }{6 }
 
  \right ]  \right \} .
 
  \right ]  \right \} .
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Here  $  \mu _ {i} $
 
Here  $  \mu _ {i} $
are the central moments of the distribution, while  $  x  ^ {[} i] = x( x - 1) \dots ( x - i + 1) $.
+
are the central moments of the distribution, while  $  x  ^ {[i]} = x( x - 1) \dots ( x - i + 1) $.
  
 
Gram–Charlier series were obtained by J.P. Gram {{Cite|G}} and C.V.L. Charlier {{Cite|Ch}} in their study of functions of the form
 
Gram–Charlier series were obtained by J.P. Gram {{Cite|G}} and C.V.L. Charlier {{Cite|Ch}} in their study of functions of the form
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$$
 
$$
  
These are convenient for the interpolation between the values  $  B ( m) = ( n!/m! ( n - m)!) p  ^ {m} q  ^ {n-} m $
+
These are convenient for the interpolation between the values  $  B ( m) = ( n!/m! ( n - m)!) p  ^ {m} q  ^ {n-m} $
 
of the general term of the [[Binomial distribution|binomial distribution]], where
 
of the general term of the [[Binomial distribution|binomial distribution]], where
  

Latest revision as of 11:31, 21 March 2022


2010 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A series defined by the expression

$$ \tag{1 } f _ {A} ( x) = \ f ( x) + \sum _ {k = 3 } ^ { n } a _ {k} f ^ { ( k) } ( x) $$

or

$$ \tag{2 } f _ {B} ( x) = \ \psi ( x) \sum _ {m = 0 } ^ { n } b _ {m} g _ {m} ( x), $$

where $ x $ is the normalized value of a random variable.

The series (1) is known as the Gram–Charlier series of type $ A $; here

$$ f ( x) = \ { \frac{1}{\sqrt {2 \pi }} } e ^ {- x ^ {2} /2 } , $$

$ f ^ { ( k) } $ is the $ k $-th derivative of $ f $, which can be represented as

$$ f ^ { ( k) } ( x) = \ (- 1) ^ {k} H _ {k} ( x) f ( x), $$

where $ H _ {k} ( x) $ are the Chebyshev–Hermite polynomials. The derivatives $ f ^ { ( k) } $ and the polynomials $ H _ {k} $ are orthogonal, owing to which the coefficients $ a _ {k} $ can be defined by the basic moments $ r _ {k} $ of the given distribution series. If one restricts to the first few terms of the series (1), one obtains

$$ f _ {A} ( x) = \ f ( x) + \frac{r _ {3} }{3!} f ^ { ( 3) } ( x) + $$

$$ + \frac{r _ {4} - 3 }{4! } f ^ { ( 4) } ( x) - \frac{r _ {5} - 10r _ {3} }{5! } f ^ { ( 3) } ( x) + \frac{r _ {4} - 15r _ {4} + 30 }{6! } f ^ { ( 6) } ( x). $$

The series (2) is known as a Gram–Charlier series of type $ B $; here

$$ \psi ( x) = \ \frac{\lambda ^ {x} }{x!} e ^ {- \lambda } ,\ \ x = 0, 1 \dots $$

while $ g _ {m} ( x) $ are polynomials analogous to the polynomials $ H _ {k} ( x) $.

If one restricts to the first terms of the series (2), one obtains

$$ f _ {B} ( x) = \ \frac{\lambda ^ {x} }{x! } e ^ {- \lambda } \left \{ 1 + \frac{\mu _ {2} - \lambda }{\lambda ^ {2} } \left [ \frac{x ^ {[2]} }{2 } - \lambda x ^ {[1]} + \frac{\lambda ^ {2} }{2 } \right ] \right . + $$

$$ + \left . \frac{\mu _ {3} - 3 \mu _ {2} + 2 \lambda }{\lambda ^ {3} } \left [ \frac{x ^ {[3]} }{6 } - { \frac \lambda {2} } x ^ {[2]} + \frac{\lambda ^ {2} }{2 } x ^ {[1]} - \frac{\lambda ^ {3} }{6 } \right ] \right \} . $$

Here $ \mu _ {i} $ are the central moments of the distribution, while $ x ^ {[i]} = x( x - 1) \dots ( x - i + 1) $.

Gram–Charlier series were obtained by J.P. Gram [G] and C.V.L. Charlier [Ch] in their study of functions of the form

$$ B _ {0} ( x) = \ { \frac{1}{2 \pi } } \int\limits _ {- \pi } ^ { {+ } \pi } e ^ {- itx } \phi ( t) dt. $$

These are convenient for the interpolation between the values $ B ( m) = ( n!/m! ( n - m)!) p ^ {m} q ^ {n-m} $ of the general term of the binomial distribution, where

$$ \phi ( t) = \ ( pe ^ {it} + q) ^ {n} = \ \sum _ {m = 0 } ^ { n } B ( m) e ^ {itm} $$

is the characteristic function of the binomial distribution. The expansion of $ \mathop{\rm ln} \phi ( t) $ in powers of $ t $ yields a Gram–Charlier series of type $ A $ for $ B _ {0} ( x) $, whereas the expansion of $ \mathop{\rm ln} \phi ( t) $ in powers of $ p $ yields a Gram–Charlier series of type $ B $.

References

[G] J.P. Gram, "Ueber die Entwicklung reeller Funktionen in Reihen mittelst der Methode der kleinsten Quadraten" J. Reine Angew. Math. , 94 (1883) pp. 41–73
[Ch] C.V.L. Charlier, "Frequency curves of type in heterograde statistics" Ark. Mat. Astr. Fysik , 9 : 25 (1914) pp. 1–17
[M] A.K. Mitropol'skii, "Curves of distributions" , Leningrad (1960) (In Russian)

Comments

Cf. also Edgeworth series.

References

[Cr] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) pp. Sect. 17.6
How to Cite This Entry:
Gram-Charlier series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gram-Charlier_series&oldid=52263
This article was adapted from an original article by A.K. Mitropol'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article