Jtdylktuy

Consider the Gumbel distributions $(P_vartheta)_{varthetaintheta}=(G(beta,mu))_{(beta,mu)in(0,infty)timesmathbb{R}}$ with distribution functions

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

Consider the productmodel $(mathbb{R}^n, mathcal{B}(mathbb{R})^{otimes n}, (P_vartheta^{otimes n})_{varthetaintheta})$

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

Can $(G(tildebeta,mu)^{otimes n})_{muinmathbb{R}}$ be written as an exponential family for a given $tildebeta$?

I found the answer on my own:

So for a given $tildebeta$ my attempt is following:

$$f_mu(x)=frac{1}{tildebeta}e^{-frac{1}{tildebeta}(x-mu)}e^{-e^{-frac{1}{tildebeta}(x-mu)}}$$

Therefore we get

begin{align}
f_mu^{otimes n}(x)=prod_{i=1}^n f_mu(x_i)&=frac{1}{tildebeta^n}expBig(-frac{1}{tildebeta}cdotsum_{i=1}^nx_iBig)expBig(frac{n}{tildebeta}muBig)expBig(-sum_{i=1}^ne^{-frac{1}{tildebeta}(x_i-mu)}Big)\
&=frac{1}{tildebeta^n}expBig(-frac{1}{tildebeta}cdotsum_{i=1}^nx_iBig)expBig(frac{n}{tildebeta}muBig)expBig(-e^{frac{mu}{tildebeta}}sum_{i=1}^ne^{-frac{x_i}{tildebeta}}Big)\
&=expBig(-e^{frac{mu}{tildebeta}}cdotsum_{i=1}^ne^{-frac{x_i}{tildebeta}}+frac{n}{tildebeta}muBig)cdotfrac{1}{tildebeta^n}expBig(-frac{1}{tildebeta}cdotsum_{i=1}^nx_iBig),
end{align}

which is an exponential family.

For given $tildemuinmathbb{R}$ one finds again

begin{align}
f_mu^{otimes n}(x)=prod_{i=1}^n f_mu(x_i)&=frac{1}{beta^n}expBig(-frac{1}{beta}cdotsum_{i=1}^nx_iBig)expBig(frac{n}{beta}tildemuBig)expBig(-e^{frac{tildemu}{beta}}sum_{i=1}^ne^{-frac{x_i}{beta}}Big)\
end{align}
The critical point is that $$forall iin{1,ldots, n}foralltext{ functions }eta_i,T_i: e^{-frac{x_i}{beta}}ne eta_i(beta)T_i(x_i)$$
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.