Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d. random variables (i.e. a random sample) with pdf:
f_X(x) = e^{-(x-\theta)} \, e^{-e^{-(x-\theta)}} · \mathbf{1}_{x\in \mathbf{R}}
The goal is to find the distribution of $T = \sum_{i=1}^n e^{-X_i}$ and also to compute $\textbf{E}(\log T)$ and $\textbf{V}(\log T)$.
Some thoughts:
I think I have found the distribution of $T$ by applying the transformation $Y = e^{-X}$. If I am not wrong, it is quite easy to see that $Y \sim \textrm{Exponential}(e^{\theta})$. Therefore, $T = \sum_{i=1}^n Y_i \sim \textrm{Gamma} (n, 1/e^{\theta})$.
However, I am unable to find a reasonable way to compute $\textbf{E}(\log T)$ or $\textbf{V}(\log T)$. The formula for the expectation of a function of a random variable leads to a very complicated integral and the only alternative I can think of, which is yet another transformation, is even worse!