Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 1 / 19H (Statistics)

Minimizing $\Vert Y-X\theta {\Vert }^2$ gives the normal equations $X^{T}X\hat{\theta }=X^{T}Y$. Full column rank makes $X^{T}X$ invertible, soWith the hat matrix $H=X(X^{T}X{)}^{-1}{X}^T$, the residual is $\hat{\epsilon }=(I-H)\epsilon$. The symmetric projection $I-H$ has rank $n-p$, henceand the latter estimator is unbiased.

Withthe model has

Let $D=Z_1+Z_2+Z_3+Z_4-2\pi$ be the observed closure error. Solving the normal equations gives

Every residual equals $D/4$, so the residual sum of squares is $D^2/4$. There is one residual degree of freedom, and therefore