View source for Gamma-correlation

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

<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>

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>

<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" />;

<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>

<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>

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

<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>

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]]]):

<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>

<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>

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]]].

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Johnson,   S. Kotz,   "Distributions in statistics" , '''2. Continuous multivariate distributions''' , Wiley  (1972)</TD></TR></table>