Uniformly Most Powerful Test and Rejection Region of Poisson Distribution 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.
 A: *

*To construct the UMP test, we have to construct the corresponding MP test. Hence, the LR is given by 
\begin{align}
\frac{L_{1}(X_1,..,X_n|\lambda_1)}{L_{1}(X_1,..,X_n|\lambda_0)} = \frac{\prod_{i=1}^n \frac{e^{-\lambda_1} \lambda_1^{x_i}}{x_i!}}{\prod_{i=1}^n \frac{e^{-\lambda_0} \lambda_0^{x_i}}{x_i!}} &= \exp\{n(\lambda_0 - \lambda_1)\}\left(\frac{\lambda_1}{\lambda_0}\right)^{\sum_i^nx_i}\\
&= \exp\{n(\lambda_0 - \lambda_1)\}\left(\frac{\lambda_1}{\lambda_0}\right)^{n\bar{x}_n} > c\,.
\end{align}
This statistic depends on the distribution of $\bar{X}_n$, hence for large enough $n$ we can use the CLT to approximate the rejection region. Note that the MP is $\Psi(\mathrm{X}) = \mathcal{I}\left( \bar{X}_n >c' \right)$.
2. 


\begin{align}
\alpha = \mathbb{E}_{\lambda_0}\Psi(\mathrm{X}) &= \mathbb{P}_{\lambda_0}\left( \bar{X}_n >c' \right)\\
&a\approx 1-\phi\left(\frac{c'-\lambda_0}{\sqrt{\lambda_0/n}} \right)\\
&c' = \lambda_0 + Z_{1-\alpha}\sqrt{\lambda_0/n}.
\end{align} 
For $\lambda_0 = 1$, 
$$
c' = 1 + Z_{1-\alpha}\sqrt{1/n}\,.
$$


*Power function for the parametric space $\Lambda = \mathbb{R}^+$.


\begin{align}
\pi(\Psi(\mathrm{X})|\Lambda) &= \mathbb{E}_{\Lambda}\Psi(\mathrm{X}) = \mathbb{P}_{\Lambda}(\bar{X}_n>c')\\
&=1-\phi\left( \frac{c'-\lambda}{\sqrt{\lambda/n}} \right),&  \forall \lambda \in \Lambda.
\end{align}
