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I have to find the uniformly most powerful size $\alpha$ test for the following pdf- $$f(x|\theta) = exp\{-(x-\theta)\}$$ $x>\theta$

The pdf is 0 for $x\leq\theta$ , Null hypothesis is $\theta = \theta_0$ and alternate is $\theta =\theta_1$. I am given an i.i.d. sample from the above pdf. I was trying to apply the Neyman pearson lemma but the relative likelihood is constant so I don’t know how to find the UMP size $\alpha$ test, can someone please provide the solution.

user601297
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  • If $\theta_1>\theta_0$, see https://math.stackexchange.com/questions/2899114/mp-test-construction-for-shifted-exponential-distribution?noredirect=1&lq=1 alongwith https://en.wikipedia.org/wiki/Uniformly_most_powerful_test#The_Karlin%E2%80%93Rubin_theorem. If $\theta_1\ne \theta_0$, there is a bit more work. – StubbornAtom Nov 03 '19 at 21:51
  • Got it, thanks a lot – user601297 Nov 03 '19 at 21:59
  • So is it $\theta_1>\theta_0$? Actually you can ignore the wiki link in my last comment. Also please choose where you want to ask instead of cross-posting. – StubbornAtom Nov 03 '19 at 22:06
  • Yes it is $\theta_1>\theta_0$, and I will keep it in mind to not cross post in the future. – user601297 Nov 03 '19 at 22:12

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