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If $X_1, X_2$ and $X_3$ are i.i.d Gamma($\theta$, 1) random variables, and $\underline{\mathbf{Y}}$ = ($X_1$ + $X_3$, $X_2$ + $X_3$) then what will be the joint density function of $\underline{\mathbf{Y}}$?

I started by defining three new random variables as $$Z_1 = X_1 + X_3,\ Z_2 = X_2 + X_3,\ Z_3 = X_3$$ and found the joint density function of $(Z_1, Z_2, Z_3)$; i.e. $f(z_1,z_2,z_3)$, where $z_1 \gt 0, z_2 \gt 0$, and $0 \lt z_3 \lt \min\{z_1, z_2\}$. After that, I obtained the joint density of $Z_1$ and $Z_2$ as $f(z_1,z_2)$ by integrating out $z_3$ over its given support. But I am unable to solve the integral, which is $$\int_{0}^{\min\{z_1, z_2\}} e^{z_3} (1 - z_3/z_1)^{\theta - 1} (1 - z_3/z_2)^{\theta - 1} z_3^{\theta - 1} dz_3.$$ Can anyone please help me in simplifying this integral?

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    $\begingroup$ It's going to be messy. (By working out special cases $\theta=1,2,3,$ etc. it will become apparent no general closed form is possible.) Could you tell us why you need a formula for the joint PDF? Perhaps there will be other ways to address whatever problem has led you to attempt this computation. $\endgroup$ – whuber Apr 16 '15 at 16:50
  • $\begingroup$ I want to check for the multivariate likelihood ratio ordering between two vectors, for which I need to find the joint density between them so that I can compare the 2 vectors. $\endgroup$ – User123 Apr 16 '15 at 17:12
  • $\begingroup$ That sounds like a practical problem where numerical approximation ought to be a feasible option. $\endgroup$ – whuber Apr 16 '15 at 17:13
  • $\begingroup$ ya i need to find this joint density for this....can anyone plz help me in solving this integral $\endgroup$ – User123 Apr 17 '15 at 3:37
  • $\begingroup$ I'm still trying to ascertain, on behalf of all interested readers, precisely what form of "solution" you need. I have suggested that you are asking for something that is not generally possible and that you should be more specific about what will work for you. $\endgroup$ – whuber Apr 17 '15 at 15:38
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The construction $(X_1 + X_3, X_2 + X_3)$ is known as Cheriyan's bivariate Gamma distribution. See the papers by:

  • Cheriyan, K.C. (1941), A bivariate correlated gamma-type distribution function, Journal of the Indian Mathematical Society, 5, 133-144

  • Ramabhadran, V.R. (1951), A multivariate gamma-type distribution, Sankhya, 11, 45-46.

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Eagleson (1964) and Mardia (1970) provide an expansion in terms of Laguerre polynomials:

  • Eagleson, G.K. (1964), Polynomial expansions of bivariate distributions, Annals of Mathematical Statistics, 35, 1208-1215.

  • Mardia, (1970), Families of Bivariate Distributions, Griffin, London.

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Szantai (1986) gives an explicit expression for the joint pdf, though don't expect pretty:

  • Szantai, T. (1986), Evaluation of special multivariate gamma distribution function, Mathematical Programming Study, 27, 1-16.
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