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?


1 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.


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