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Let $X_1, \dots, X_n \sim \mathrm{Poi}(\lambda)$. I know that

(a) There exists a UMP test of $H_0: \lambda = 0.5 $ vs. $H_1: \lambda = 1.5$

(b) There exists a UMP test of $H_0: \lambda \leq 1$ vs. $H_1: \lambda > 1$.

How can I show that the UMP test of $H_0: \lambda = 1$ vs. $H_1: \lambda \neq 1$ does not exist?

My attempt: I solved part (a) and (b) using Neyman-Pearson lemma and Karlin-Rubin theorem. I think the problem has something to do with the necessity part of Neyman-Pearson lemma, but not sure how to write a clear argument.

967723
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    Suppose a UMP test exists. Then it should be UMP for both the alternatives $\lambda>1$ and $\lambda<1$. Now argue that UMP tests for the one-sided alternatives cannot be exactly same and arrive at a contradiction. This is true for any distribution of the regular exponential family. – StubbornAtom Dec 09 '19 at 10:08
  • @StubbornAtom Could you explain what do you mean by UMP tests for one-sided alternatives ($\lambda = 1$ vs. $\lambda \neq 1$?) cannot be exactly the same? – 967723 Dec 09 '19 at 14:20
  • What does the alternatives $\lambda > 1$ and $\lambda < 1$ mean? – 967723 Dec 09 '19 at 16:59
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    Find UMP tests for the cases $H_1: \lambda>1$ and $H_1:\lambda <1$ for the same $H_0:\lambda=1$. – StubbornAtom Dec 09 '19 at 17:37

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