Expression of Kibble's bivariate Gamma distribution PDF

Question

I'm studying the Kibble's bivariate Gamma distribution and found one inconsistency between different papers and I'm not sure which is correct.

In the Smith et al. 1982, A bivariate Gamma Probability Distribution with Application to Gust Model, the 4-parameter PDF of bivarate Gamma given in Eq. 2.5 is:

$f(t_1, t_2; \gamma, \rho) = \frac{(t_1 t_2)^{(\gamma - 1) / 2} exp[-(t_1+t_2)/(1-\rho)]}{\rho^{(\gamma-1)/2}(1-\rho)\Gamma(\gamma)} I_{\gamma-1}[\frac{2 (\rho t_1 t_2)^{1/2}}{1-\rho}]$

because $t_1 = \beta_1 x$, $t_2 = \beta_2 y$, this has a $(\beta_1 \beta_2)^{(\gamma - 1)/2}$ term when replacing $t_1$ and $t_2$ with $x$ and $y$.

However, in Downton 1970 Bivariate Exponential Distributions in Reliability Theory (because Smith et al. 1982 mentioned that this function "as derived by Kibble (1941) and reported by Downton 1970", and I didn't manage to find the Kibble 1941 paper), they have in Eq. 2.12:

$f_v(t_1,t_2) = \frac{(\mu_1 \mu_2)^v}{(1-\rho)\Gamma(v)} (\frac{t_1 t_2}{\rho \mu_1 \mu_2})^{(v-1)/2} exp(-\frac{\mu_1 t_1 + \mu_2 t_2}{1-\rho}) I_{v-1}[\frac{2\sqrt{\rho \mu_1 \mu_2 t_1 t_2}}{1 - \rho}]$

so the product of scale parameters is raised to the power of $(v+1)/2$.

This same equation also appears in Iliopoulos, George and Karlis, Dimitris and Ntzoufras, Ioannis, 2005: Bayesian Estimation in Kibble's Bivariate Gamma Distribution, Eq. 1.

And in Izawa, T., 1965: Two or Multi-dimensional Gamma-type Distribution and Its Application to Rainfall Data, the power of scale parameters is also $(v+1)/2$.

I'm confused about this inconsistency. Could anyone give some help?

Answer 1

You need to include the determinant of the Jacobian of the transformation from $t_1$ and $t_2$ to $x$ and $y$. (See equation (12.2) and associated explanations here).

Thus, the joint pdf in terms of $x$ and $y$ is,

\begin{align*} g(x,y;\gamma,\rho,\beta_1,\beta_2) &= f(\beta_1 x, \beta_2 y; \gamma,\rho) \begin{vmatrix}\dfrac{\partial t_1}{\partial x} & \dfrac{\partial t_1}{\partial y} \\ \dfrac{\partial t_2}{\partial x} & \dfrac{\partial t_2}{\partial y} \end{vmatrix}\\ &= f(\beta_1 x, \beta_2 y; \gamma,\rho) \begin{vmatrix}\beta_1 & 0 \\ 0 & \beta_2 \end{vmatrix}\\ &= f(\beta_1 x, \beta_2 y; \gamma,\rho) \beta_1 \beta_2. \end{align*}

This extra factor $\beta_1 \beta_2$ explains why the product of scale parameters should be raised to the power $(\gamma+1)/2$ and not $(\gamma-1)/2$.

Note: I have the Kibble (1941) paper and can confirm that he does not work with scale parameters at all.