Gamma-correlation

The two-dimensional distribution of non-negative random dependent variables $X_1$ and $X_2$ defined by the density

$$p(x_1,x_2)=p_1(x_1)p_2(x_2)\sum _{k=0}^{\mathrm{\infty }}{c}_k{a}_k{L}_k^{{\alpha }_1}(x_1)L_k^{{\alpha }_2}(x_2),$$

where

$$0\le x_{\nu }<\mathrm{\infty };{\alpha }_{\nu }\ge \gamma >-1;{\rho }_{\nu }(x_{\nu })=x_{\nu }^{{\alpha }_{\nu }}\frac{e^{-x_{\nu }}}{\mathrm{\Gamma }({\alpha }_{\nu }+1)},$$

$L_k^{{\alpha }_{\nu }}(x_{\nu })$ are the Laguerre polynomials, orthonormalized on the positive semi-axis with weight $p_{\nu }(x_{\nu })$, $\nu =1,2$;

$$a_k=\frac{\mathrm{\Gamma }(\gamma +k+1)}{\mathrm{\Gamma }(\gamma +1)}\sqrt{\frac{\mathrm{\Gamma }({\alpha }_1+1)\mathrm{\Gamma }({\alpha }_2+1)}{\mathrm{\Gamma }({\alpha }_1+k+1)\mathrm{\Gamma }({\alpha }_2+k+1)}};$$

$$c_k=\underset{0}{\overset{1}{\int }}{\lambda }^{k}dF(\lambda ),k=0,1,\dots ;$$

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

$$\varphi (t_1,t_2)=\underset{0}{\overset{1}{\int }}\frac{dF(\lambda )}{[1-i{t}_1-i{t}_2-t_1{t}_2(1-\lambda ){]}^{1+\gamma }}.$$

If $F(\lambda )$ is such that $\mathsf{P}\{\lambda =R\}=1$, then $c_k=R^k$, $\varphi (t_1,t_2)=[1-i{t}_1-i{t}_2-t_1{t}_2(1-R){]}^{-1-\gamma }$, and $R$ is the correlation coefficient between $X_1$ and $X_2$( $0\le R\le 1$). In this last case the density series can be summed using the formula (cf. [2]):

$$p(x_1,x_2)=\frac{(x_1{x}_2{)}^{\gamma }{e}^{-x_1-x_2}}{{\mathrm{\Gamma }}^2(\gamma +1)}\sum _{k=0}^{\mathrm{\infty }}{R}^k{L}_k^{\gamma }(x_1)L_k^{\gamma }(x_2)=$$

$$=\frac{e^{-(x_1+x_2)/(1-R)}}{(1-R)\mathrm{\Gamma }(\gamma +1)}{\left(\frac{x_1{x}_2}{R}\right)}^{\gamma /2}{I}_{\gamma }\left(\frac{2\sqrt{x_1{x}_{2}R}}{1-R}\right),$$

where $I_{\gamma }$ is the Bessel function of an imaginary argument [2].

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Comments[edit]

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.

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