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A series defined by the expression
 
A series defined by the expression
  
<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/g/g044/g044730/g0447301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
f _ {A} ( x)  = \
 +
f ( x) +
 +
\sum _ {k = 3 } ^ { n }
 +
a _ {k} f ^ { ( k) } ( x)
 +
$$
  
 
or
 
or
  
<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/g/g044/g044730/g0447302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
f _ {B} ( x)  = \
 +
\psi ( x)
 +
\sum _ {m = 0 } ^ { n }
 +
b _ {m} g _ {m} ( x),
 +
$$
 +
 
 +
where  $  x $
 +
is the normalized value of a random variable.
 +
 
 +
The series (1) is known as the Gram–Charlier series of type  $  A $;
 +
here
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g0447303.png" /> is the normalized value of a random variable.
+
$$
 +
f ( x)  = \
 +
{
 +
\frac{1}{\sqrt {2 \pi }}
 +
}
 +
e ^ {- x  ^ {2} /2 } ,
 +
$$
  
The series (1) is known as the Gram–Charlier series of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g0447305.png" />; here
+
$  f ^ { ( k) } $
 +
is the $  k $-
 +
th derivative of $  f $,
 +
which can be represented as
  
<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/g/g044/g044730/g0447306.png" /></td> </tr></table>
+
$$
 +
f ^ { ( k) } ( x)  = \
 +
(- 1)  ^ {k} H _ {k} ( x) f ( x),
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g0447307.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g0447308.png" />-th derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g0447309.png" />, which can be represented as
+
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
  
<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/g/g044/g044730/g04473010.png" /></td> </tr></table>
+
$$
 +
f _ {A} ( x)  = \
 +
f ( x) +
 +
\frac{r _ {3} }{3!}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473011.png" /> are the Chebyshev–Hermite polynomials. The derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473012.png" /> and the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473013.png" /> are orthogonal, owing to which the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473014.png" /> can be defined by the basic moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473015.png" /> of the given distribution series. If one restricts to the first few terms of the series (1), one obtains
+
f ^ { ( 3) } ( x) +
 +
$$
  
<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/g/g044/g044730/g04473016.png" /></td> </tr></table>
+
$$
 +
+
  
<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/g/g044/g044730/g04473017.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473019.png" />; here
+
The series (2) is known as a Gram–Charlier series of type $  B $;  
 +
here
  
<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/g/g044/g044730/g04473020.png" /></td> </tr></table>
+
$$
 +
\psi ( x)  = \
  
while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473021.png" /> are polynomials analogous to the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473022.png" />.
+
\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
 
If one restricts to the first terms of the series (2), one obtains
  
<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/g/g044/g044730/g04473023.png" /></td> </tr></table>
+
$$
 +
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 .
  
<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/g/g044/g044730/g04473024.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473025.png" /> are the central moments of the distribution, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473026.png" />.
+
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 {{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
  
<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/g/g044/g044730/g04473027.png" /></td> </tr></table>
+
$$
 +
B _ {0} ( x)  = \
 +
{
 +
\frac{1}{2 \pi }
 +
}
 +
\int\limits _ {- \pi } ^ { {+ }  \pi }
 +
e ^ {- itx } \phi ( t)  dt.
 +
$$
  
These are convenient for the interpolation between the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473028.png" /> of the general term of the [[Binomial distribution|binomial distribution]], where
+
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
  
<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/g/g044/g044730/g04473029.png" /></td> </tr></table>
+
$$
 +
\phi ( t)  = \
 +
( pe  ^ {it} + q)  ^ {n}  = \
 +
\sum _ {m = 0 } ^ { n }
 +
B ( m) e  ^ {itm}
 +
$$
  
is the characteristic function of the binomial distribution. The expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473030.png" /> in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473031.png" /> yields a Gram–Charlier series of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473033.png" />, whereas the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473034.png" /> in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473035.png" /> yields a Gram–Charlier series of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044730/g04473036.png" />.
+
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====
 
====References====

Revision as of 19:42, 5 June 2020


2020 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=47112
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