Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 18H (Statistics)

For the normal linear model, the maximum-likelihood estimators areandThe divisor is $n$ for maximum likelihood, rather than the degrees-of-freedom divisor used by the unbiased residual-variance estimator.

Letbe the hat matrix. Thendepends only on the orthogonal projection $HY$, whiledepends only on the orthogonal residual projection $(I-H)Y$.

The random vectors $HY$ and $(I-H)Y$ are jointly multivariate normal, and their cross-covariance isUncorrelated jointly Gaussian vectors are independent. Therefore

Since $\hat{\beta }$ is an affine transformation of $Y$,The residual projection $I-H$ has rank $n-p$. Cochran's theorem therefore gives

Under $H_0$, write the singular value decompositionThenThe coordinates of $U^T\epsilon /\sigma$ are independent standard normal variables. Henceindependently. By parts (ii) and (iii),is independent of all the $W_j$. Consequently

For arbitrary $\beta$,Independence from part (ii) and $Z=n{\hat{\sigma }}^2/{\sigma }^2\sim {\chi }_{n-p}^2$ giveBecause $n-p>2$,Thus