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Consider the Gumbel distributions $(P_\vartheta)_{\vartheta\in\theta}=(G(\beta,\mu))_{(\beta,\mu)\in(0,\infty)\times\mathbb{R}}$ with distribution functions

$$F_{\beta,\mu}(x)=e^{-e^{-\frac{1}{\beta}(x-\mu)}}$$

Consider the productmodel $(\mathbb{R}^n, \mathcal{B}(\mathbb{R})^{\otimes n}, (P_\vartheta^{\otimes n})_{\vartheta\in\theta})$

Can $(G(\beta,\tilde\mu)^{\otimes n})_{\beta\in(0,\infty)}$ be written as an exponential family for a given $\tilde\mu$?

Can $(G(\tilde\beta,\mu)^{\otimes n})_{\mu\in\mathbb{R}}$ be written as an exponential family for a given $\tilde\beta$?

I found the answer on my own:

So for a given $\tilde\beta$ my attempt is following:

$$f_\mu(x)=\frac{1}{\tilde\beta}e^{-\frac{1}{\tilde\beta}(x-\mu)}e^{-e^{-\frac{1}{\tilde\beta}(x-\mu)}}$$

Therefore we get

\begin{align} f_\mu^{\otimes n}(x)=\prod_{i=1}^n f_\mu(x_i)&=\frac{1}{\tilde\beta^n}\exp\Big(-\frac{1}{\tilde\beta}\cdot\sum_{i=1}^nx_i\Big)\exp\Big(\frac{n}{\tilde\beta}\mu\Big)\exp\Big(-\sum_{i=1}^ne^{-\frac{1}{\tilde\beta}(x_i-\mu)}\Big)\\ &=\frac{1}{\tilde\beta^n}\exp\Big(-\frac{1}{\tilde\beta}\cdot\sum_{i=1}^nx_i\Big)\exp\Big(\frac{n}{\tilde\beta}\mu\Big)\exp\Big(-e^{\frac{\mu}{\tilde\beta}}\sum_{i=1}^ne^{-\frac{x_i}{\tilde\beta}}\Big)\\ &=\exp\Big(-e^{\frac{\mu}{\tilde\beta}}\cdot\sum_{i=1}^ne^{-\frac{x_i}{\tilde\beta}}+\frac{n}{\tilde\beta}\mu\Big)\cdot\frac{1}{\tilde\beta^n}\exp\Big(-\frac{1}{\tilde\beta}\cdot\sum_{i=1}^nx_i\Big), \end{align}

which is an exponential family.

For given $\tilde\mu\in\mathbb{R}$ one finds again

\begin{align} f_\mu^{\otimes n}(x)=\prod_{i=1}^n f_\mu(x_i)&=\frac{1}{\beta^n}\exp\Big(-\frac{1}{\beta}\cdot\sum_{i=1}^nx_i\Big)\exp\Big(\frac{n}{\beta}\tilde\mu\Big)\exp\Big(-e^{\frac{\tilde\mu}{\beta}}\sum_{i=1}^ne^{-\frac{x_i}{\beta}}\Big)\\ \end{align} The critical point is that $$\forall i\in\{1,\ldots, n\}\forall\text{ functions }\eta_i,T_i: e^{-\frac{x_i}{\beta}}\ne \eta_i(\beta)T_i(x_i)$$ Therefore this is not an exponential family, because one would need to to separate the variables $\beta$ and $x_i$ in the expression

$$e^{-\frac{x_i}{\beta}}$$

to write down an exponential family.

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