Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 4 / Solution

Letand define the functional principal component scoresUse empirical centered scores ${\hat{\xi }}_{ik},{\hat{\eta }}_{il}$ from the sample covariance eigenfunctions and formThe finite-dimensional statistic is

Under $H_0$, $Y=\epsilon$ is independent of $X$. The population cross-covariances vanish, andThe multivariate central limit theorem, consistency of the empirical eigenpairs, and Slutsky theorem therefore giveRejecting above the $(1-\alpha )$ quantile gives an asymptotic level-$\alpha$ test.

Under a fixed alternative, $\mathbb{E}[{\eta }_l{\xi }_k]$ is the $(l,k)$ coordinate of the cross-covariance operator $C_{YX}=B{C}_X$, where $B$ is the Hilbert-Schmidt operator with kernel $\beta$. ThusThe test is consistent whenever this retained block contains a nonzero cross-covariance; fixed truncation can miss alternatives outside the selected principal-component subspaces.