Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 3 / d

Since $X_j\sim \mathrm{Poisson}(n_j{\lambda }_j)$, the rate MLE has asymptotic variance ${\lambda }_j/n_j$. The delta method for the logarithm givesIndependence of the arms therefore yieldsUnder $H_0:{\lambda }_T={\lambda }_C=\lambda$, the asymptotic variance of the log rate ratio is