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Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$.

(1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of hypotheses: $H_0: \lambda\leq1$ versus $H_A: \lambda >1$.

$\bf{My \ thoughts:}$ I know I can use Karlin-Rubin to help me with this part of the problem. Using Factorization Theorem, I get that $\sum^n_{i=1}X_i$ is a sufficient and complete statistic. Checking the MLR and using Karlin-Rubin,I get $\alpha = P_{\lambda} \Bigl(\sum^n_{i=1}X_i > 1 \Bigr).$

(2) Using the CLT, provide an expression for the rejection region for this test.

$\bf{My \ thoughts:}$ Using the CLT, I know I need to start by finding the asymptotic distribution of $\sum^n_{i=1}X_i$ and go from there. Just not sure on setting that up.

(3) Find the power function for this test.

$\bf{My \ thoughts:}$ This doesn't seem like it would be too difficult and should come from having what I need from 1 and 2.

Any help is greatly appreciated.

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This question seems more suitable for Cross-Validated (you might have better luck there). Just an FYI. –  AlanH Mar 11 '13 at 17:32
    
Thanks, I will try that. –  user45185 Mar 12 '13 at 5:32
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