Namespaces
Variants
Actions

Difference between revisions of "Semi-invariant"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A numerical characteristic of random variables related to the concept of a [[Moment|moment]] of higher order. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s0841901.png" /> is a random vector, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s0841902.png" /> is its characteristic function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s0841903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s0841904.png" />,
+
<!--
 +
s0841901.png
 +
$#A+1 = 45 n = 0
 +
$#C+1 = 45 : ~/encyclopedia/old_files/data/S084/S.0804190 Semi\AAhinvariant
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/s/s084/s084190/s0841905.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
and if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s0841906.png" /> the moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s0841907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s0841908.png" />, then the (mixed) moments
+
A numerical characteristic of random variables related to the concept of a [[Moment|moment]] of higher order. If  $  \xi = ( \xi _ {1} \dots \xi _ {k} ) $
 +
is a random vector,  $  \phi _  \xi  ( t) = {\mathsf E} e ^ {i ( t, \xi ) } $
 +
is its characteristic function, $  t = ( t _ {1} \dots t _ {k} ) $,
 +
$  t _ {i} \in \mathbf R $,
  
<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/s/s084/s084190/s0841909.png" /></td> </tr></table>
+
$$
 +
( t, \xi )  = \
 +
\sum _ {i = 1 } ^ { k }
 +
t _ {i} \xi _ {i} ,
 +
$$
  
exist for all non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419011.png" />. Under these conditions,
+
and if for some  $  n \geq  1 $
 +
the moments  $  {\mathsf E} | \xi _ {i} |  ^ {n} < \infty $,
 +
$  i = 1 \dots k $,  
 +
then the (mixed) moments
  
<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/s/s084/s084190/s08419012.png" /></td> </tr></table>
+
$$
 +
m _  \xi  ^ {( \nu _ {1} \dots \nu _ {k} ) }  = \
 +
{\mathsf E} \xi _ {1} ^ {\nu _ {1} }
 +
{} \dots \xi _ {k} ^ {\nu _ {k} }
 +
$$
  
<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/s/s084/s084190/s08419013.png" /></td> </tr></table>
+
exist for all non-negative integers  $  \nu _ {1} \dots \nu _ {k} $
 +
such that  $  \nu _ {1} + \dots + \nu _ {k} \leq  n $.  
 +
Under these conditions,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419014.png" />, and for sufficiently small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419015.png" /> the principal value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419016.png" /> can be represented by Taylor's formula as
+
$$
 +
\phi _  \xi  ( t)  = \
 +
\sum _ {\nu _ {1} + \dots + \nu _ {k} \leq  n }
  
<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/s/s084/s084190/s08419017.png" /></td> </tr></table>
+
\frac{i ^ {\nu _ {1} + \dots + \nu _ {k} } }{\nu _ {1} ! \dots \nu _ {k} ! }
  
<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/s/s084/s084190/s08419018.png" /></td> </tr></table>
+
m _  \xi  ^ {( \nu _ {1} \dots \nu _ {k} ) } \times
 +
$$
  
where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419019.png" /> are called the (mixed) semi-invariants, or cumulants, of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419020.png" /> of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419021.png" />. For independent random vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419023.png" />,
+
$$
 +
\times
 +
t _ {1} ^ {\nu _ {1} } \dots t _ {k} ^ {\nu _ {k} } + o (| t |  ^ {n} ),
 +
$$
  
<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/s/s084/s084190/s08419024.png" /></td> </tr></table>
+
where  $  | t | = | t _ {1} | + \dots + | t _ {k} | $,
 +
and for sufficiently small  $  | t | $
 +
the principal value of  $  \mathop{\rm ln}  \phi _  \xi  ( t) $
 +
can be represented by Taylor's formula as
 +
 
 +
$$
 +
\mathop{\rm ln}  \phi _  \xi  ( t)  = \
 +
\sum _ {\nu _ {1} + \dots + \nu _ {k} \leq  n }
 +
 
 +
\frac{i ^ {\nu _ {1} + \dots + \nu _ {k} } }{\nu _ {1} ! \dots \nu _ {k} ! }
 +
 
 +
s _  \xi  ^ {( \nu _ {1} \dots \nu _ {k} ) } \times
 +
$$
 +
 
 +
$$
 +
\times
 +
t _ {1} ^ {\nu _ {1} } \dots t _ {k} ^ {\nu _ {k} } + o (| t |  ^ {n} ),
 +
$$
 +
 
 +
where the coefficients  $  s _  \xi  ^ {( \nu _ {1} \dots \nu _ {k} ) } $
 +
are called the (mixed) semi-invariants, or cumulants, of order  $  \nu = ( \nu _ {1} \dots \nu _ {k} ) $
 +
of the vector  $  \xi = ( \xi _ {1} \dots \xi _ {k} ) $.
 +
For independent random vectors  $  \xi = ( \xi _ {1} \dots \xi _ {k} ) $
 +
and  $  \eta = ( \eta _ {1} \dots \eta _ {k} ) $,
 +
 
 +
$$
 +
s _ {\xi + \eta }  ^ {( \nu _ {1} \dots \nu _ {k} ) }  = \
 +
s _  \xi  ^ {( \nu _ {1} \dots \nu _ {k} ) } +
 +
s _  \eta  ^ {( \nu _ {1} \dots \nu _ {k} ) } ,
 +
$$
  
 
that is, the semi-invariant of a sum of independent random vectors is the sum of their semi-invariants. This is the reason for the term  "semi-invariant" , which reflects the additive property of independent variables (but, in general, the property does not hold for dependent variables).
 
that is, the semi-invariant of a sum of independent random vectors is the sum of their semi-invariants. This is the reason for the term  "semi-invariant" , which reflects the additive property of independent variables (but, in general, the property does not hold for dependent variables).
Line 27: Line 86:
 
The following formulas, connecting moments and semi-invariants, hold:
 
The following formulas, connecting moments and semi-invariants, hold:
  
<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/s/s084/s084190/s08419025.png" /></td> </tr></table>
+
$$
 +
m _  \xi  ^ {( \nu ) }  = \
 +
\sum  ^ {*}
 +
{
 +
\frac{1}{q!}
 +
}
 +
 
 +
\frac{\nu ! }{\lambda  ^ {(} 1) ! \dots \lambda  ^ {(} q) ! }
 +
 
 +
\prod _ {p = 1 } ^ { q }
 +
s _  \xi  ^ {( \lambda  ^ {(} p) ) } ,
 +
$$
 +
 
 +
$$
 +
s _  \xi  ^ {( \nu ) }  = \sum  ^ {*}
 +
\frac{(- 1) ^ {q - 1
 +
} }{q }
 +
 +
\frac{\nu ! }{\lambda  ^ {(} 1) ! \dots \lambda  ^ {(} q) ! }
  
<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/s/s084/s084190/s08419026.png" /></td> </tr></table>
+
\prod _ {p = 1 } ^ { q }  m _  \xi  ^ {( \lambda  ^ {(} p) ) } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419027.png" /> denotes summation over all ordered sets of non-negative integer vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419029.png" />, with as sum the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419030.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419031.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419032.png" />, and similarly for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419033.png" />.) In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419034.png" /> is a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084190/s08419037.png" />, then
+
where $  \sum  ^ {*} $
 +
denotes summation over all ordered sets of non-negative integer vectors $  \lambda  ^ {(} p) $,
 +
$  | \lambda  ^ {(} p) | > 0 $,  
 +
with as sum the vector $  \nu $.  
 +
(Here $  v! $
 +
is defined as $  v ! = v _ {1} ! \dots v _ {k} ! $,  
 +
and similarly for the $  \lambda  ^ {(} p) ! $.)  
 +
In particular, if $  \xi $
 +
is a random variable $  ( k = 1) $,  
 +
$  m _ {n} = m _  \xi  ^ {(} n) = {\mathsf E} \xi  ^ {n} $,  
 +
and $  s _ {n} = s _  \xi  ^ {(} n) $,  
 +
then
  
<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/s/s084/s084190/s08419038.png" /></td> </tr></table>
+
$$
 +
m _ {1}  = s _ {1} ,
 +
$$
  
<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/s/s084/s084190/s08419039.png" /></td> </tr></table>
+
$$
 +
m _ {2}  = s _ {2} + s _ {1}  ^ {2} ,
 +
$$
  
<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/s/s084/s084190/s08419040.png" /></td> </tr></table>
+
$$
 +
m _ {3}  = s _ {3} + 3s _ {1} s _ {2} + s _ {1}  ^ {3} ,
 +
$$
  
<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/s/s084/s084190/s08419041.png" /></td> </tr></table>
+
$$
 +
m _ {4}  = s _ {4} + 3s _ {2}  ^ {2} + 4s _ {1} s _ {3} + 6s _ {1}  ^ {2} s _ {2} + s _ {1}  ^ {4} ,
 +
$$
  
 
and
 
and
  
<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/s/s084/s084190/s08419042.png" /></td> </tr></table>
+
$$
 +
s _ {1}  = m _ {1}  (= {\mathsf E} \xi ),
 +
$$
  
<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/s/s084/s084190/s08419043.png" /></td> </tr></table>
+
$$
 +
s _ {2}  = m _ {2} - m _ {1}  ^ {2}  (= {\mathsf D} \xi ),
 +
$$
  
<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/s/s084/s084190/s08419044.png" /></td> </tr></table>
+
$$
 +
s _ {3}  = m _ {3} - 3m _ {1} m _ {2} + 2m _ {1}  ^ {3} ,
 +
$$
  
<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/s/s084/s084190/s08419045.png" /></td> </tr></table>
+
$$
 +
s _ {4}  = m _ {4} - 3m _ {2}  ^ {2} - 4m _ {1} m _ {3} + 12m _ {1}  ^ {2} m _ {2} - 6m _ {1}  ^ {4} .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.P. Leonov,  A.N. Shiryaev,  "On a method of calculation of semi-invariants"  ''Theory Probab. Appl.'' , '''4''' :  3  (1959)  pp. 319–329  ''Teor. Veroyatnost. i Primen.'' , '''4''' :  3  (1959)  pp. 342–355</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Shiryaev,  "Probability" , Springer  (1984)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.P. Leonov,  A.N. Shiryaev,  "On a method of calculation of semi-invariants"  ''Theory Probab. Appl.'' , '''4''' :  3  (1959)  pp. 319–329  ''Teor. Veroyatnost. i Primen.'' , '''4''' :  3  (1959)  pp. 342–355</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Shiryaev,  "Probability" , Springer  (1984)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Stuart,  J.K. Ord,  "Kendall's advanced theory of statistics" , Griffin  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Schmetterer,  "Introduction to mathematical statistics" , Springer  (1974)  pp. Chapt. 1, §42  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Rényi,  "Probability theory" , North-Holland  (1970)  pp. Chapt. 3, §15</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Stuart,  J.K. Ord,  "Kendall's advanced theory of statistics" , Griffin  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Schmetterer,  "Introduction to mathematical statistics" , Springer  (1974)  pp. Chapt. 1, §42  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Rényi,  "Probability theory" , North-Holland  (1970)  pp. Chapt. 3, §15</TD></TR></table>

Revision as of 08:13, 6 June 2020


A numerical characteristic of random variables related to the concept of a moment of higher order. If $ \xi = ( \xi _ {1} \dots \xi _ {k} ) $ is a random vector, $ \phi _ \xi ( t) = {\mathsf E} e ^ {i ( t, \xi ) } $ is its characteristic function, $ t = ( t _ {1} \dots t _ {k} ) $, $ t _ {i} \in \mathbf R $,

$$ ( t, \xi ) = \ \sum _ {i = 1 } ^ { k } t _ {i} \xi _ {i} , $$

and if for some $ n \geq 1 $ the moments $ {\mathsf E} | \xi _ {i} | ^ {n} < \infty $, $ i = 1 \dots k $, then the (mixed) moments

$$ m _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } = \ {\mathsf E} \xi _ {1} ^ {\nu _ {1} } {} \dots \xi _ {k} ^ {\nu _ {k} } $$

exist for all non-negative integers $ \nu _ {1} \dots \nu _ {k} $ such that $ \nu _ {1} + \dots + \nu _ {k} \leq n $. Under these conditions,

$$ \phi _ \xi ( t) = \ \sum _ {\nu _ {1} + \dots + \nu _ {k} \leq n } \frac{i ^ {\nu _ {1} + \dots + \nu _ {k} } }{\nu _ {1} ! \dots \nu _ {k} ! } m _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } \times $$

$$ \times t _ {1} ^ {\nu _ {1} } \dots t _ {k} ^ {\nu _ {k} } + o (| t | ^ {n} ), $$

where $ | t | = | t _ {1} | + \dots + | t _ {k} | $, and for sufficiently small $ | t | $ the principal value of $ \mathop{\rm ln} \phi _ \xi ( t) $ can be represented by Taylor's formula as

$$ \mathop{\rm ln} \phi _ \xi ( t) = \ \sum _ {\nu _ {1} + \dots + \nu _ {k} \leq n } \frac{i ^ {\nu _ {1} + \dots + \nu _ {k} } }{\nu _ {1} ! \dots \nu _ {k} ! } s _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } \times $$

$$ \times t _ {1} ^ {\nu _ {1} } \dots t _ {k} ^ {\nu _ {k} } + o (| t | ^ {n} ), $$

where the coefficients $ s _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } $ are called the (mixed) semi-invariants, or cumulants, of order $ \nu = ( \nu _ {1} \dots \nu _ {k} ) $ of the vector $ \xi = ( \xi _ {1} \dots \xi _ {k} ) $. For independent random vectors $ \xi = ( \xi _ {1} \dots \xi _ {k} ) $ and $ \eta = ( \eta _ {1} \dots \eta _ {k} ) $,

$$ s _ {\xi + \eta } ^ {( \nu _ {1} \dots \nu _ {k} ) } = \ s _ \xi ^ {( \nu _ {1} \dots \nu _ {k} ) } + s _ \eta ^ {( \nu _ {1} \dots \nu _ {k} ) } , $$

that is, the semi-invariant of a sum of independent random vectors is the sum of their semi-invariants. This is the reason for the term "semi-invariant" , which reflects the additive property of independent variables (but, in general, the property does not hold for dependent variables).

The following formulas, connecting moments and semi-invariants, hold:

$$ m _ \xi ^ {( \nu ) } = \ \sum ^ {*} { \frac{1}{q!} } \frac{\nu ! }{\lambda ^ {(} 1) ! \dots \lambda ^ {(} q) ! } \prod _ {p = 1 } ^ { q } s _ \xi ^ {( \lambda ^ {(} p) ) } , $$

$$ s _ \xi ^ {( \nu ) } = \sum ^ {*} \frac{(- 1) ^ {q - 1 } }{q } \frac{\nu ! }{\lambda ^ {(} 1) ! \dots \lambda ^ {(} q) ! } \prod _ {p = 1 } ^ { q } m _ \xi ^ {( \lambda ^ {(} p) ) } , $$

where $ \sum ^ {*} $ denotes summation over all ordered sets of non-negative integer vectors $ \lambda ^ {(} p) $, $ | \lambda ^ {(} p) | > 0 $, with as sum the vector $ \nu $. (Here $ v! $ is defined as $ v ! = v _ {1} ! \dots v _ {k} ! $, and similarly for the $ \lambda ^ {(} p) ! $.) In particular, if $ \xi $ is a random variable $ ( k = 1) $, $ m _ {n} = m _ \xi ^ {(} n) = {\mathsf E} \xi ^ {n} $, and $ s _ {n} = s _ \xi ^ {(} n) $, then

$$ m _ {1} = s _ {1} , $$

$$ m _ {2} = s _ {2} + s _ {1} ^ {2} , $$

$$ m _ {3} = s _ {3} + 3s _ {1} s _ {2} + s _ {1} ^ {3} , $$

$$ m _ {4} = s _ {4} + 3s _ {2} ^ {2} + 4s _ {1} s _ {3} + 6s _ {1} ^ {2} s _ {2} + s _ {1} ^ {4} , $$

and

$$ s _ {1} = m _ {1} (= {\mathsf E} \xi ), $$

$$ s _ {2} = m _ {2} - m _ {1} ^ {2} (= {\mathsf D} \xi ), $$

$$ s _ {3} = m _ {3} - 3m _ {1} m _ {2} + 2m _ {1} ^ {3} , $$

$$ s _ {4} = m _ {4} - 3m _ {2} ^ {2} - 4m _ {1} m _ {3} + 12m _ {1} ^ {2} m _ {2} - 6m _ {1} ^ {4} . $$

References

[1] V.P. Leonov, A.N. Shiryaev, "On a method of calculation of semi-invariants" Theory Probab. Appl. , 4 : 3 (1959) pp. 319–329 Teor. Veroyatnost. i Primen. , 4 : 3 (1959) pp. 342–355
[2] A.N. Shiryaev, "Probability" , Springer (1984) (Translated from Russian)

Comments

References

[a1] A. Stuart, J.K. Ord, "Kendall's advanced theory of statistics" , Griffin (1987)
[a2] L. Schmetterer, "Introduction to mathematical statistics" , Springer (1974) pp. Chapt. 1, §42 (Translated from German)
[a3] A. Rényi, "Probability theory" , North-Holland (1970) pp. Chapt. 3, §15
How to Cite This Entry:
Semi-invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-invariant&oldid=48662
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article