Let $X_1$, $X_2$ be two independent random variables with different gamma distributions, and $X = \max\{X_1, X_2\}$.
Are there known results for the distribution of $X$? Actually I only need to know $\mathrm E[X]$ (exact value or approximation).
In case there are no known results for that, I would still find it useful to consider the particular case that the two distributions have the same scale (or rate) parameter (but different shape parameters).
The Wikipedia entry for order statistics give some analytic results for the probability density function for various order statistics.
Given $X_1, X_2 \sim \operatorname{Gamma}(\alpha, \beta)$, and define $X_{(2)} = \max\left\{ X_1, X_2 \right\}$. Then, the distribution for $X_{(2)}$ is
$$ f_{X_{(2)}}(x)=\dfrac{\Gamma(3)}{\Gamma(2)} f_X(x) F_X(x)$$
Where $f_X$ is the density for the gamma distribution and $F_X$ is the cumulative distribution function. This should be fairly easy to simulate in R to
x <- replicate(100000, {
max(rgamma(2, 2, 1))
})
hist(x, probability = TRUE, ylim = c(0, 0.5))
curve(
gamma(3) / gamma(2) * dgamma(x, 2, 1)*(pgamma(x, 2, 1)), from = 0,
to = max(x), add=TRUE
)
Now consider the case where
$$ X_1 \sim \operatorname{Gamma}(\alpha, \beta) $$ $$ X_2 \sim \operatorname{Gamma}(\alpha ^\prime, \beta^\prime) $$
and again define $X_{(2)} = \max\left\{ X_1, X_2 \right\}$ where $X_1 \perp X_2$.
This means we can write the joint CDF as
$$ F_{X_{(2)}}(x) = \Pr(X_1 \lt x \cap X_2 \lt x) = F_{X_1}(x)F_{X_2}(x)$$
The joint density is then obtained by differentiating the above expression to yield
$$ f_{X_{(2)} } = f_{X_1}(x)F_{X_2}(x) + F_{X_1}(x)f_{X_2}(x) $$
Again, we can simulate this in R
x <- replicate(100000, {
x1 <- rgamma(1, 2, 1)
x2 <- rgamma(1, 2, 3)
max(c(x1, x2))
})
hist(x, probability = TRUE, ylim = c(0, 0.5), breaks = 100)
curve(
dgamma(x, 2, 1)*pgamma(x, 2, 3) + pgamma(x, 2, 1)*dgamma(x, 2, 3),
from = 0, to = max(x), add=TRUE
)
