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The two-dimensional distribution of non-negative random dependent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g0432901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g0432902.png" /> defined by the density
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<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/g043/g043290/g0432903.png" /></td> </tr></table>
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The two-dimensional distribution of non-negative random dependent variables  $  X _ {1} $
 +
and  $  X _ {2} $
 +
defined by the density
 +
 
 +
$$
 +
p ( x _ {1} , x _ {2} )  = \
 +
p _ {1} ( x _ {1} )  p _ {2} ( x _ {2} )
 +
\sum _ {k = 0 } ^  \infty 
 +
c _ {k} a _ {k} L _ {k} ^ {\alpha _ {1} }
 +
( x _ {1} ) L _ {k} ^ {\alpha _ {2} } ( x _ {2} ),
 +
$$
  
 
where
 
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/g043/g043290/g0432904.png" /></td> </tr></table>
+
$$
 +
0  \leq  x _  \nu  < \infty ; \ \
 +
\alpha _  \nu  \geq  \gamma  > - 1; \ \
 +
\rho _  \nu  ( x _  \nu  )  = \
 +
x _  \nu  ^ {\alpha _  \nu  }
 +
 
 +
\frac{e ^ {- x _  \nu  } }{\Gamma ( \alpha _  \nu  + 1) }
 +
,
 +
$$
 +
 
 +
$  L _ {k} ^ {\alpha _  \nu  } ( x _  \nu  ) $
 +
are the [[Laguerre polynomials|Laguerre polynomials]], orthonormalized on the positive semi-axis with weight  $  p _  \nu  ( x _  \nu  ) $,
 +
$  \nu = 1, 2 $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g0432905.png" /> are the [[Laguerre polynomials|Laguerre polynomials]], orthonormalized on the positive semi-axis with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g0432906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g0432907.png" />;
+
$$
 +
a _ {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/g/g043/g043290/g0432908.png" /></td> </tr></table>
+
\frac{\Gamma ( \gamma + k + 1) }{\Gamma ( \gamma + 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/g/g043/g043290/g0432909.png" /></td> </tr></table>
+
\sqrt {
 +
\frac{\Gamma ( \alpha _ {1} + 1) \Gamma ( \alpha _ {2} + 1) }{\Gamma ( \alpha _ {1} + k + 1) \Gamma ( \alpha _ {2} + k + 1) }
 +
} ;
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329010.png" /> is an arbitrary distribution function on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329011.png" />. The correlation coefficient between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329013.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329015.png" />, a symmetric gamma-correlation is obtained; in such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329017.png" /> and the form of the corresponding characteristic function is
+
$$
 +
c _ {k}  = \int\limits _ { 0 } ^ { 1 }  \lambda  ^ {k}  dF ( \lambda ),\  k = 0, 1 ,\dots;
 +
$$
  
<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/g043/g043290/g04329018.png" /></td> </tr></table>
+
and  $  F ( \lambda ) $
 +
is an arbitrary distribution function on the segment  $  [ 0, 1] $.
 +
The correlation coefficient between  $  X _ {1} $
 +
and  $  X _ {2} $
 +
is  $  c _ {1} { {( \gamma + 1) } / \sqrt {( \alpha _ {1} + 1 )( \alpha _ {2} + 1 ) } } $.
 +
If  $  \alpha _ {1} = \alpha _ {2} = \gamma $,
 +
a symmetric gamma-correlation is obtained; in such a case  $  a _ {k} = 1 $,
 +
$  k = 0, 1 \dots $
 +
and the form of the corresponding characteristic function is
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329019.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329023.png" /> is the [[Correlation coefficient|correlation coefficient]] between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329026.png" />). In this last case the density series can be summed using the formula (cf. [[#References|[2]]]):
+
$$
 +
\phi ( t _ {1} , t _ {2} )  = \
 +
\int\limits _ { 0 } ^ { 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/g/g043/g043290/g04329027.png" /></td> </tr></table>
+
\frac{dF ( \lambda ) }{[ 1 - it _ {1} - it _ {2} - t _ {1} t _ {2} ( 1 - \lambda )] ^ {1 + \gamma } }
 +
.
 +
$$
  
<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/g043/g043290/g04329028.png" /></td> </tr></table>
+
If  $  F ( \lambda ) $
 +
is such that  $  {\mathsf P} \{ \lambda = R \} = 1 $,
 +
then  $  c _ {k} = R  ^ {k} $,
 +
$  \phi ( t _ {1} , t _ {2} ) = [ 1 - it _ {1} - it _ {2} - t _ {1} t _ {2} ( 1 - R)] ^ {- 1- \gamma } $,
 +
and  $  R $
 +
is the [[Correlation coefficient|correlation coefficient]] between  $  X _ {1} $
 +
and  $  X _ {2} $(
 +
0 \leq  R \leq  1 $).  
 +
In this last case the density series can be summed using the formula (cf. [[#References|[2]]]):
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043290/g04329029.png" /> is the Bessel function of an imaginary argument [[#References|[2]]].
+
$$
 +
p ( x _ {1} , x _ {2} )  = \
 +
 
 +
\frac{( x _ {1} x _ {2} )  ^  \gamma  e ^ {- x _ {1} - x _ {2} } }{\Gamma  ^ {2} ( \gamma + 1) }
 +
 
 +
\sum _ {k = 0 } ^  \infty 
 +
R  ^ {k} L _ {k}  ^  \gamma  ( x _ {1} ) L _ {k}  ^  \gamma  ( x _ {2} ) =
 +
$$
 +
 
 +
$$
 +
= \
 +
 
 +
\frac{e ^ {- ( x _ {1} + x _ {2} ) / ( 1 - R) } }{( 1 - R) \Gamma
 +
( \gamma + 1) }
 +
\left (
 +
\frac{x _ {1} x _ {2} }{R }
 +
\right ) ^ {\gamma / 2 } I _  \gamma  \left (
 +
\frac{2 \sqrt
 +
{x _ {1} x _ {2} R } }{1 - R }
 +
\right ) ,
 +
$$
 +
 
 +
where  $  I _  \gamma  $
 +
is the Bessel function of an imaginary argument [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.O. Sarmanov,  ''Trudy Gidrologichesk. Inst.'' , '''162'''  (1969)  pp. 37–61</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Myller-Lebedeff,  "Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen"  ''Math. Ann.'' , '''64'''  (1907)  pp. 388–416</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.O. Sarmanov,  ''Trudy Gidrologichesk. Inst.'' , '''162'''  (1969)  pp. 37–61</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Myller-Lebedeff,  "Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen"  ''Math. Ann.'' , '''64'''  (1907)  pp. 388–416</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:41, 5 June 2020


The two-dimensional distribution of non-negative random dependent variables $ X _ {1} $ and $ X _ {2} $ defined by the density

$$ p ( x _ {1} , x _ {2} ) = \ p _ {1} ( x _ {1} ) p _ {2} ( x _ {2} ) \sum _ {k = 0 } ^ \infty c _ {k} a _ {k} L _ {k} ^ {\alpha _ {1} } ( x _ {1} ) L _ {k} ^ {\alpha _ {2} } ( x _ {2} ), $$

where

$$ 0 \leq x _ \nu < \infty ; \ \ \alpha _ \nu \geq \gamma > - 1; \ \ \rho _ \nu ( x _ \nu ) = \ x _ \nu ^ {\alpha _ \nu } \frac{e ^ {- x _ \nu } }{\Gamma ( \alpha _ \nu + 1) } , $$

$ L _ {k} ^ {\alpha _ \nu } ( x _ \nu ) $ are the Laguerre polynomials, orthonormalized on the positive semi-axis with weight $ p _ \nu ( x _ \nu ) $, $ \nu = 1, 2 $;

$$ a _ {k} = \ \frac{\Gamma ( \gamma + k + 1) }{\Gamma ( \gamma + 1) } \sqrt { \frac{\Gamma ( \alpha _ {1} + 1) \Gamma ( \alpha _ {2} + 1) }{\Gamma ( \alpha _ {1} + k + 1) \Gamma ( \alpha _ {2} + k + 1) } } ; $$

$$ c _ {k} = \int\limits _ { 0 } ^ { 1 } \lambda ^ {k} dF ( \lambda ),\ k = 0, 1 ,\dots; $$

and $ F ( \lambda ) $ is an arbitrary distribution function on the segment $ [ 0, 1] $. The correlation coefficient between $ X _ {1} $ and $ X _ {2} $ is $ c _ {1} { {( \gamma + 1) } / \sqrt {( \alpha _ {1} + 1 )( \alpha _ {2} + 1 ) } } $. If $ \alpha _ {1} = \alpha _ {2} = \gamma $, a symmetric gamma-correlation is obtained; in such a case $ a _ {k} = 1 $, $ k = 0, 1 \dots $ and the form of the corresponding characteristic function is

$$ \phi ( t _ {1} , t _ {2} ) = \ \int\limits _ { 0 } ^ { 1 } \frac{dF ( \lambda ) }{[ 1 - it _ {1} - it _ {2} - t _ {1} t _ {2} ( 1 - \lambda )] ^ {1 + \gamma } } . $$

If $ F ( \lambda ) $ is such that $ {\mathsf P} \{ \lambda = R \} = 1 $, then $ c _ {k} = R ^ {k} $, $ \phi ( t _ {1} , t _ {2} ) = [ 1 - it _ {1} - it _ {2} - t _ {1} t _ {2} ( 1 - R)] ^ {- 1- \gamma } $, and $ R $ is the correlation coefficient between $ X _ {1} $ and $ X _ {2} $( $ 0 \leq R \leq 1 $). In this last case the density series can be summed using the formula (cf. [2]):

$$ p ( x _ {1} , x _ {2} ) = \ \frac{( x _ {1} x _ {2} ) ^ \gamma e ^ {- x _ {1} - x _ {2} } }{\Gamma ^ {2} ( \gamma + 1) } \sum _ {k = 0 } ^ \infty R ^ {k} L _ {k} ^ \gamma ( x _ {1} ) L _ {k} ^ \gamma ( x _ {2} ) = $$

$$ = \ \frac{e ^ {- ( x _ {1} + x _ {2} ) / ( 1 - R) } }{( 1 - R) \Gamma ( \gamma + 1) } \left ( \frac{x _ {1} x _ {2} }{R } \right ) ^ {\gamma / 2 } I _ \gamma \left ( \frac{2 \sqrt {x _ {1} x _ {2} R } }{1 - R } \right ) , $$

where $ I _ \gamma $ is the Bessel function of an imaginary argument [2].

References

[1] I.O. Sarmanov, Trudy Gidrologichesk. Inst. , 162 (1969) pp. 37–61
[2] W. Myller-Lebedeff, "Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen" Math. Ann. , 64 (1907) pp. 388–416

Comments

This bivariate distribution is just one of the many possible multivariate generalizations of the (univariate) gamma-distribution. See [a1], Chapt. 40 for a survey as well as more details on this one.

References

[a1] N.L. Johnson, S. Kotz, "Distributions in statistics" , 2. Continuous multivariate distributions , Wiley (1972)
How to Cite This Entry:
Gamma-correlation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gamma-correlation&oldid=47037
This article was adapted from an original article by O.V. Sarmanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article