Thanks for the answer, those minus signs got the worst of me.. To follow up, this means that the canonical link function is $b(\mu )^{-1}=(-\sqrt{-2\mu})^{-1}$ @Xi'an ?

Why delete the comments @Xi'an ?

How did you pull out the minus sign @Xi'an form the square root operation? Sorry it's been a long night maybe I oversee something basic here..

Agreed, the minus sign is a typo and that' probably why I got it wrong all along. Thanks alot @Xi'an !

I think it now makes sense for $b(\theta)=\sqrt{2\theta}$

Oh man @Xi'an of course $𝜂(\mu)=\frac{1}{2\mu^2}$. I will specify the support in the question right now.

Btw @Glen_b are you familiar with canonical exponential families? I'm struggling with a pdf I' trying to rewrite in canonical form posted here maybe you could help me out.

Thank you so much @Glen_b I think I'm actually starting to get it now!

No @Glen_b that topic is new to me. I have just dealt with test statistics up to this point. Could you perhaps share a link?

I left with finding $P(|\bar{X}_n-5|>C_{a,n})=a$, where $\bar{X}_n$ is the expected value equal to $5\lambda$ and here is where I'm stuck. Can you please help me out @Glen_b ?

Hi @Glen_b thank you so much for your comments! You said that the "kernel is $𝑥$ to a power times $𝑒^{−𝑥/𝑐}$", and I'm still trying to figure out $\alpha$ and $\beta$. Although according to that info then $\alpha=5$and $\beta=\lambda$ but that $\beta$ seems off since the test does not depend on the estimate of $𝜆$. Any input on this?

Yes @Glen_b wikipedia is definitely bookmarked. Any suggestion on how I should tackle this one? If you need any clarification on something that I might have missed like for the other question let me know :)

Btw, I'm struggling with an hypothesis test I posted here. Do you mind having a look whenever you have some spare time @Glen_b ?