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I was looking at a problem in the book of "Statistical Inference" second edition by George Casella and Roger L. Berger from chapter 6 that deals with sufficient statistics, minimal sufficient statistics, complete statistics, etc.

In problem 6.20 part d) I have the following question

Let $X_1,X_2,\cdots X_n$ be iid observations with the following pdf:

$f(x|\theta)=e^{-(x-\theta)}\cdot exp(-e^{-(x-\theta)})$

Where $-\infty <x< \infty$ and $-\infty <\theta< \infty$

Find a complete sufficient statistic or show that one does not exist.

I proved that this belongs to the exponential family rewriting the pdf the following way: $f(x|\theta)=e^{-x}e^{\theta}\cdot exp(-e^{-x}\cdot e^{\theta})$

$h(x)=e^{-x}$; $c(\theta)=e^{\theta}$; $w(\theta)=e^{\theta}$; $t(x)=-e^{-x}$

so I concluded that it has a complete sufficient statistic due to the theorem:

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However, I then looked at a solution manual and they suggested a different answer so I'm kind of confused and don't know what I did wrong.

This was the solution:

enter image description here

Any help is much appreciated, thank you in advance.

Yeison Augusto Quiceno Duran
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    The solution is non-sensical if indeed the support of the density is the entire real line. The sufficient statistic is then the sum of the $\exp\{X_i\}$'s. Imho, the [solution manual](http://www.ams.sunysb.edu/~zhu/ams570/Solutions-Casella-Berger.pdf) does not correspond to the same edition and hence 6.20 is not the same problem in different editions. – Xi'an Feb 01 '22 at 17:27
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    You are looking at the Gumbel distribution. Related question is: https://stats.stackexchange.com/questions/276391 – Sextus Empiricus Feb 01 '22 at 17:36
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    I checked both the second edition and the solution manual for that edition and there is clearly a mistake in the answer. Maybe the person who wrote the answer got confused with the location-scale Gumbel, which stands outside the exponential families... – Xi'an Feb 01 '22 at 17:43
  • Ok, thanks. I was starting to wonder if I was not worthy of living anymore haha. Now that I have verified that this distribution indeed belongs to the exponential family I won't be depriving myself of any more sleep. Thanks again. – Yeison Augusto Quiceno Duran Feb 01 '22 at 17:47

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