Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / b / v / Solution

Ignoring factors that do not depend on $\theta$, each observed side-effect contributes $\theta e^{-\theta x_i}$ and each censored observation contributes $e^{-\theta x_i}$. ThusThe score equation $v_{+}/\theta -x_{+}=0$ givesThis event-count divided by person-time estimator is the sample analogue of the expectation ratio in part iv.