ratio distribution of gamma with different parameter

Question

Let $X$ be gamma distributed random variable with parameters $a$ and $b$. Let $W$ be gamma distributed random variable with parameters $c$ and $d$, such that \begin{equation} f_X(x) = \frac{x^{a-1}e^{-\frac{x}{b}}}{b^a\Gamma(a)}\quad \text{and} \quad f_W(w) = \frac{w^{c-1}e^{-\frac{w}{d}}}{d^c\Gamma(c)}. \end{equation}

The problem is to find the distribution of $\frac{X}{W+s}$ where $s$ is a constant. So far I have got, Let $Z = \frac{X}{W+s} = \frac{X}{Y}$, such that the CDF of $Z$ now can be stated as. \begin{equation} \int_s^{\infty}\int_{0}^{yz}\frac{x^{a-1}e^{\frac{-x}{b}}(y-s)^{(c-1)}e^{\frac{-(y-s)}{d}}}{b^a\Gamma(a)d^c\Gamma(c)}\mathrm{d}x\mathrm{d}y \end{equation}

Integrating the following will yield in \begin{equation} 1-\int_{s}^{\infty}\frac{\Gamma_{inc}(a,\frac{yz}{b})(y-s)^{(c-1)}e^{-\frac{y-s}{d}}}{\Gamma(a)\Gamma(c)d^c}\mathrm{d}y \end{equation} where $\Gamma_{inc}$ is the incomplete gamma function.

I could not solve the integral term. Is there any way to solve it, or even any other way the problem of finding the distribution of $Z$.

Answer 1

pdf of .... Ratio of Gamma to shifted Gamma

Let $X \sim \text{Gamma}(a,b)$ with pdf $f(x)$:

Let $W \sim \text{Gamma}(c,d)$ be independent of $X$ and define parameter $s>0$. Then, $W+s$ has a shifted Gamma distribution. Then, by any of the usual methods, the pdf of $Y = \frac{\large1}{\large W+s}$ is $g(y)$:

We seek the pdf of $Z = \frac{\large X}{\large W+s} = X*Y$. As this is just the product of two independent random variables, ... the solution pdf, say $h(z)$ is given immediately by:

where I am using the TransformProduct function from the mathStatica package for Mathematica to do the nitty-gritties.

All done.

Monte Carlo check

It's always a good idea to check symbolic output. Here is a quick Monte Carlo simulation of the 'empirical' pdf of $Z = \frac{\large X}{\large W+s}$ (the jagged BLUE curve) compared to the theoretical solution $h(z)$ (the RED curve) derived above, when:

 params = {a -> 2.3, b -> 3, c -> 4, d -> 5, s -> 7}

Looks fine.

Notes

1. For detail on the Hypergeometric1F1Regularized function, see: http://reference.wolfram.com/language/ref/Hypergeometric1F1Regularized.html

2. As disclosure, I should perhaps add that I am one of the authors of the software used above.

3. The solution above works for real values of parameter $a$ (but not for integer values). The problem with integer values is some odd manifestation of the way Mathematica simplifies or converts certain Gamma functions into trig funcs, if I recall correctly.