Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 18H (Statistics)

The log likelihood, up to constants, isDifferentiation gives the generalized least squares estimatorand maximizing over the scale givesThe divisor $n$ is appropriate for maximum likelihood rather than unbiased estimation.

The estimator is an affine transformation of the multivariate normal distribution. Its mean is $\beta$, and direct covariance calculation gives

Write $\hat{\beta }=AY$, whereAny other linear unbiased estimator is $CY$ with $CX=I$, so $C=A+D$ and $DX=0$. The cross covariance vanishes becauseThereforeBy the Gauss-Markov theorem,

Let $\nu =n/2-p$. Under $H_0$, the two residual sums of squares are independent and satisfyHenceUnder $H_1$, the numerator is scaled by the smaller variance, so small values provide evidence against the null. If $F_{\nu ,\nu }^{-1}(\alpha )$ denotes the lower $\alpha$ quantile, a size-$\alpha$ test rejects when