# Uniformly Most Powerful (UMP) statistic for N Poisson distributions

For samples of $$M$$ Poisson-distributed datapoints $$X_{1, r},...,X_{M,r}$$ $$\sim$$ $$Pois(\lambda_{r})$$, and $$N$$ such distributions ($$1 < r < N$$), I have defined a likelihood function to describe my data as:

$$$$\mathcal{L}\left(data \mid H_{}\right)=\prod_{r=1}^{N} \prod_{i=1}^{M} \frac{e^{-\lambda_{r}} \lambda_{r}^{X_{i, r}}}{X_{i, r} !}$$$$

I am attempting to find the uniformly most powerful (UMP) statistic using the Neyman-Pearson lemma, to test for the hypothesis $$H_{a}: \lambda_{r} = \lambda_{a, r}, \lambda_{a, r} > \lambda_{0, r} \ \forall \ r \in (1, 2, .. N)$$ against the null $$H_{0}: \lambda_{r} = \lambda_{0, r}$$. The trouble is, when I find the likelihood ratio,

$$\Lambda=\frac{\mathcal{L}\left(data \mid H_{0}\right)}{\mathcal{L}\left(data \mid H_{a}\right)}=\prod_{r=1}^{N} \frac{e^{M} \lambda_{a, r}}{e^{M} \lambda_{0, r}}\left(\frac{\lambda_{0, r}}{\lambda_{a, r}}\right)^{\sum_{i=1}^{M}X_{i, r}} \leq c$$

After taking the log of both sides, I get what appears to be a weighted combination of a Poisson-distributed variable whose distribution I am having difficulty describing:

$$\sum_{r=1}^{N} \sum_{i=1}^{M}{X_{i, r}} \log \left(\frac{\lambda_{0}, r}{\lambda_{1, r}}\right) \geq \log (c)-\sum_{r=1}^{N}\left(\lambda_{a, r}-\lambda_{0, r}\right) M$$

I recognize that the sum of Independent Poisson random variables should also be Poisson distributed: $$\sum_{r=1}^{N} \sum_{i=1}^{M}{X_{i, r}}$$

but I can't seem to deconvolve this expression to find a neat statistic. I came across a great post by whuber that describes one approach to approximate the behaviour of this statistic using Cornish-Fisher expansion, but I wonder if there is a more practical way to find the likelihood ratio test for this case?