Say I have $n$ i.i.d. random variables $\mathbf{X}_n = \{X_1,X_2,\ldots,X_n\}$, where $X_i \sim \operatorname{Erlang}(k,\theta)$ or $\Gamma(k,\theta)$.
Define $$Z_n := \max \mathbf{X}_n$$
I know that $Z_n$ is in the domain of attraction of the Gumbel distribution as shown in e.g. the answer to this question
But it seems that the sequence converges rather slowly (at least for the normalizing constants given in the above link), most likely too slow for my purposes.
I've modeled the time it takes for $n$ parallel sequences of $k$ tasks to finish as $n$ Gamma-distributed random variables. Assuming all tasks are i.i.d. exponentially distributed. And want to know the time it takes for all sequences to finish, the maximum.
My $n$'s are the sequence of powers of two, which start off very small, and therefore not that close to a denormalized Gumbel. These small powers of two are however still of interest.
I can of course use the exact distribution for some specific $n$, i.e.
$$E(Z_n) = \int_0^\infty xf_{Z_n}(x) \, \text{d} x$$
But $f_{Z_n}$ looks rather unsavory:
$$f_{Z_n}(x) = \frac{\text{d}}{\text{d}x}\left( 1 - \sum_{n=0}^{k-1} \frac{e^{-\theta x}(\theta x)^n}{n!}\right)^m\\ = \frac{\theta m e^{-\theta x}(\theta x)^{k-1}\left(1 -\frac{(k-1)!\sum_{l=0}^{k-1}\frac{e^{-\theta x}(\theta x)^l}{l!} }{k!}\right)^{m-1}}{k!}$$
(can also be expressed with gamma functions) and I can't seem to get a very clean result for the expectation of $Z_n$. I am however but an engineer and not that great at complicated integrals. This does seem to graph out a rather regular-looking curve, as seen in the question I linked earlier, so maybe this could be solved (at least numerically for specific Gamma-distributions).
It would be nice to have a general distribution, with a known expectation, which the GEV-distribution gives. The only trouble is the rather large errors to the exact distribution for small $n$.
So, my questions are:
- Is there a clean way to represent the expectation of the exact distribution of the max of $n$ Gamma distributed random variables?
- Is there a better way to pick the normalizing constants so that the normalized Gumbel converges towards the exact distribution quicker?
- Is there any information about the rate of convergence for Gumbel-distributions and their domains of attraction? Or the error as a function of $n$?
