Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 225 / 2 / d / Solution

The squared-norm test is omnibus: every fixed nonzero mean eventually changes $\Vert {\overline{X}}_n\Vert$. Its null law, however, is an infinite weighted chi-squared distribution and requires accurate estimation of enough covariance eigenvalues; noisy low-variance directions can also make calibration inefficient.

The FPCA mean test has the simple ${\chi }_K^2$ limit and standardizes retained directions by their variances. It is effective when the signal lies in the leading principal component subspace, but choosing $K$ introduces a tuning decision and truncation makes the test blind to alternatives orthogonal to that subspace. Close or repeated eigenvalues also make individual empirical eigenfunctions unstable.

The sign-flip randomization test can provide finite-sample calibration and avoids estimating a limiting covariance spectrum. Its exactness requires central symmetry, which is stronger than merely having zero mean, and exhaustive enumeration costs $2^n$ evaluations; Monte Carlo sign flips introduce simulation error. Its power still depends on the statistic used inside the randomization scheme.