 Likelihood-ratio test of area-proportional Poisson means

Two equally numerous groups of independent counts with Poisson distributions have areas $a$ and $2a$. Under common density their group means are in ratio $1:2$. With group sums $S_1,S_2$ and $T=S_1+S_2$, the maximized alternative-to-null likelihood ratio isUse zero-count limits and $\Lambda =1$ when $T=0$. The regular Wilks theorem calibration has one degree of freedom. Exactly conditioning on $T$ gives $S_1\sim \mathrm{Binomial}(T,1/3)$ under the null; this supports small-count testing without the large-sample approximation.