Gamma-correlation

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)