I need to determine the generalised likelihood ratio in order to test whether a set of values $X$ are such that each $x_{i} \in X$ are given by a common poisson distribution $Po(\lambda)$ or alternatively they each are given by their own $Po(\lambda_{i})$.
Here the former is the null hypothesis and the latter is the alternate hypothesis.
The part I'm having difficulty with is that I'm not really understanding what the generalised likelihood ratio really is. From what I understand, generally, the generalised likelihood ratio is given by $l(\theta_{0})/l(\hat\theta)$ where $\theta_{0}$ and $\hat\theta$ are MLEs.
The problem is I don't understand what in this case these likelihood functions and MLEs represent. I can follow simpler cases, like determining if a coin is biased or not, but I'm just not getting it here.
If anyone could offer insight into how to calculate the generalised likelihood ratio, or explain what $l(\theta_{0})$ and $l(\hat\theta)$ are, it would be much appreciated.
