Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 2 / c / i

For each substudy define the log-odds treatment effectThe normal random-effects log-likelihood, up to an additive constant, isIts score equations giveThese are the maximum-likelihood estimators rather than the unbiased sample-variance estimator. At an interior solution with ${\hat{\sigma }}^2>0$, the Hessian in $(\mu ,{\sigma }^2)$ is negative definite, which is the required second-order condition.