I would like to find the UMP level-$\alpha$ test for $H_0: \theta \leq \theta_0$ vs $H_1: \theta \gt \theta_0$ for poisson distribution. I know that the distribution has mlr in $\sum x_i$. By Karlin-Rubin, the UMP must have the form:
1 if $\sum x_i \gt c$
$\lambda$ if $\sum x_i = c$
o otherwise
where c, $\lambda$ satisfy: $P_{\theta_0}(\sum x_i \gt c) + \lambda P_{\theta_0}(\sum x_i = c) = \alpha$
I also know that $\sum x_i$ follows a poisson distribution with parameter $n\theta$. Where do I go from here?
Any help is appreciated