Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 5J (Statistical Modelling)

A $(1-\alpha )$ prediction interval is a pair of statistics $L(Y),U(Y)$ such that, for every $\beta$ and ${\sigma }^2>0$, an independent future response $Y^{\ast }$ at the specified input satisfies

In the normal linear model, assume $X$ has full column rank and writeFor a new observationindependent of the training sample, the prediction error isThe two terms on the right are independent centered normal random variables, whileConsequently, with the new-point regression leveragewe have

The assumption in the question givesThis variable is independent of $\hat{\beta }$; it is also independent of the new error ${\epsilon }^{\ast }$. Therefore it is independent of the standardized prediction error, and the defining normal-over-chi-square representation of Student's t-distribution givesIf $t_{n-p,1-\alpha /2}$ is the corresponding upper quantile, symmetry yieldsSolving these inequalities for $Y^{\ast }$ proves that the prediction interval in a normal linear model isIts center and endpoints depend only on the observed data and $x^{\ast }$, and the pivotal calculation proves the required coverage uniformly in the unknown $\beta$ and ${\sigma }^2$.