# Finding a sufficient statistic for an iid sample of the Gumbel distribution

$G(x;\alpha, \beta) = \exp\{-\beta e^{-\alpha x}\}$ for $x \in \mathbb{R}$ is a distribution (Gumbel family).

Side question: is $G(x;\alpha, \beta)$ a member of the exponential family? I do not think it is because the $-\beta e^{-\alpha x}$ cannot be expressed as $\eta(\alpha,\beta) \cdot T(x) - A(\alpha, \beta)$...

The question asks to find sufficient statistics for $X_1, \dots, X_n$ an iid sample from $G(x;\alpha,\beta)$.

Then, \begin{align*} P(X_1 \leq x_1, \dots, X_n \leq x_n) &= \prod_{i=1}^n P(X \leq x_i) \\ &= \prod_{i=1}^n \exp\{-\beta e^{-\alpha x}\} \\ &= \exp\left\{ -\beta\sum_{i=1}^n e^{-\alpha x_i}\right\} \end{align*} I'm not really sure where to go from here since I cannot seem to separate the $x_i$'s from the $\alpha$ and $\beta$'s... Any hints or comments would be appreciated!

• Are you sure Gumbel distribution is defined as you written above? – kaka Sep 17 '14 at 5:32
• @kaka In the problem, it is defined this way... I think it may be a variation? – symmetricuser Sep 17 '14 at 5:41