Disclaimer: This is a question I couldn't solve. I feel it is ethical to state this here so you can evaluate how much of your solution will share through either statements or hints.
Let $X_{1},…,X_n $ be an $i.i.d.$ $Poiss(\lambda)$ distributed random variable for some unknown $\lambda . 0$. Now consider the following hypothesis test:
$H_0:λ=λ_0$ v.s. $H_1:λ≠λ_0$ where $λ_0>0$.
Question: What is the asymptotic p-value?
Here's what I know:
Let $\overline{X}_n$ be the average of the random variable $X$. Thanks to the Central Limit Theorem, I can write:
$\dfrac{|\overline{X}_n - \lambda_0|}{\sigma/\sqrt{n}} \xrightarrow{(d)} N(0,1)$ as $n$ goes to infinity.
Given it's a two-sided test, according to [1] I can write
$P_{\lambda_0}\left(\dfrac{|\overline{X}_n - \lambda_0|}{\sigma/\sqrt{n}} > z_{\alpha/2}\right)$ $\rightarrow \alpha$
$P_{\lambda_0}\left(|W| > z_{\alpha/2}\right)$ $\rightarrow \alpha$
where $W$ is the Wald Test and $z_{\alpha/2} = \Phi^{-1}(1 - \alpha/2)$
Also from [1], I can write
$p-value = P_{\lambda_0}(|W| > |z_{\alpha/2}|) = 2.\Phi(-|z_{\alpha/2}|)$
This is where I'm stuck. I know all of these statements are true, however, I cannot work with the knowledge I have to obtain the answer I need. I'm still re-reading about hypothesis testing and p-values to have a better grip on problems like this.
References
[1] Wasserman, Larry. All of statistics: a concise course in statistical inference. Springer Science & Business Media, 2004, second edition.