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Let $X_1,\dots,X_n$ be a sample from $U(0,θ), θ>0$. Find UMP size-$α$ for testing $H0:θ=θ0$ against $H1:θ≠θ0$.

I worked on it by dividing the regions into two:

  1. H0:θ=θ0 against H1:θ<θ0
  2. H0:θ=θ0 against H1:θ>θ0

By using the MLR property of the statistic $\max(x_1,…,x_n)$, I got max(x1,…,xn)<θ0 α^(1/n) as the rejection region for 1st case and max(x1,…,xn)>θ0 (1-α)^(1/n) for the second case. From this, how will I reach to a UMP?

This answer is where I'm trying to reach. I am doubting my answer for the second case. How can I reach to that answer?

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Statistics aspirant
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  • This is one problem where working backwards is easier. That is, given the answer in the linked post, show that it is UMP against both the alternatives $H_A:\theta>\theta_0$ and $H_B:\theta<\theta_0$. The tests you obtain here using MLR property do not directly lead to the UMP test for testing $H_0$ against $H_1:\theta\ne\theta_0$. UMP test for $(H_0,H_1)$ is unique, but UMP tests for $(H_0,H_A)$ and $(H_0,H_B)$ are not unique. – StubbornAtom Aug 23 '21 at 20:03
  • Hii.... how do we know that there are more than one UMP test for those hypothesis? – Statistics aspirant Aug 24 '21 at 05:01
  • Through Neyman-Pearson lemma. – StubbornAtom Aug 24 '21 at 10:55

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