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I am asked to prove whether or not the following distribution (extreme value distribution) is part of the exponential family: $$ f_Y(y;\mu,\phi)=\frac{1}{\phi}\exp[-\frac{y-\mu}{\phi}+\exp(-\frac{y-\mu}{\phi})]$$ My definition of the exponential family is the following: enter image description here

I have tried rewriting the density function and obtained: $$f_Y(y;\mu,\phi)=\exp \left[\log\frac{1}{\phi}-\frac{y}{\phi}+\frac{\mu}{\phi}-\exp\left(\frac{-y+\mu}{\phi}\right)\right]$$ However, at this point I am stuck. How do I get the density function into the form of the exponential family distribution from this point? Any help would be very much appreciated!

I H
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Mikalo
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  • You can see here that this is a member of exponential family with natural parameter $\theta(\mu)=-e^{\mu/\phi}$ provided $\phi$ is known. So the answer to your question depends on $\phi$ being fixed or not. – StubbornAtom Dec 26 '21 at 15:33

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