Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 29L (Principles of Statistics)

The log-likelihood, score, and Fisher information areandFor independent identically distributed observations,

Writing the score as a column vector and differentiating under the integral,

For the simple null, the restricted maximum-likelihood estimator is $\tilde{\theta }={\theta }_0$, and henceThe individual scores are independent, have mean zero by part (ii), and have covariance $I_1({\theta }_0)$. The multivariate central limit theorem givesAfter multiplying by $I_1({\theta }_0{)}^{-1/2}$, the limit is standard normal in ${\mathbb{R}}^p$. The continuous mapping theorem therefore yields

Write $v={\sigma }^2$. For one observation, the score isandUnder the restriction $v=1$, the restricted maximum-likelihood estimator is $\tilde{\mu }=\bar{X}$. At $(\bar{X},1)$ the first score component is zero, while the second isSince $I_1(\bar{X},1{)}^{-1}=\mathrm{diag}(1,2)$, substitution into the score statistic gives

Under $H_0$, regardless of the unknown value of $\mu$,Write $Q_n$ as a sum of $n-1$ independent squared standard normals. Since such a square has mean one and variance two, the central limit theorem givesNowso Slutsky's theorem makes this converge to $N(0,1)$. Squaring and using the continuous mapping theorem yields