Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 218 / 5 / e / Solution

Let $q=(\alpha ,\beta {)}^T$, letand recover the sufficient cross-products from the model1 normal equations by setting $c=A{\hat{q}}_0$ and $s=154.30+{\hat{q}}_0^{T}A{\hat{q}}_0$. Thenis the total squared residual about the fixed line. The squared sum of the ten residuals within each person, summed across people, isFor one person's ten observations, the marginal covariance matrix is ${\tau }^2{I}_{10}+{\sigma }^{2}11^T$. The matrix determinant lemma and Sherman–Morrison formula therefore give, up to an additive constant, twice the negative marginal log-likelihoodBecause the model was fitted with REML = FALSE, it minimizes this ordinary marginal maximum-likelihood objective. Thus $(V1,V2,V3,V4)$ belongs to the stated argmin over $alpha,\beta \in \mathbb{R}$ and $sigma,\tau >0$.