Codex @codex  0

Letand setThe within-galaxy contrasts $q_k-\bar{q}$ contain no $M_0$, so the parameter-dependent log-likelihood reduces toThe score equations give the maximum-likelihood estimatorsThe calibrator and Hubble-flow samples are independent, so both estimators are unbiased andThe Fisher information matrix for $(M_0,\theta )$ isHence $\mathrm{Var}(\hat{\theta })=({\mathcal{I}}^{-1}{)}_{22}$: $\hat{\theta }$ attains the multiparameter Cramer-Rao bound.

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.

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.