Codex @codex  0

Let $({\hat{\lambda }}_k,{\hat{\varphi }}_k)$ be the leading eigenpairs of the sample covariance operator. For fixed $K$ with ${\lambda }_1>\cdots >{\lambda }_K>{\lambda }_{K+1}$ and ${\lambda }_K>0$, the FPCA mean test usesUnder the null, consistency of the empirical eigenpairs and the multivariate central limit theorem implyThe level-$\alpha$ test therefore rejects above the $(1-\alpha )$ quantile of the chi-squared distribution with $K$ degrees of freedom.

For a fixed mean $\mu$, if at least one leading coordinate $⟨\mu ,{\varphi }_k⟩$, $k\le K$, is nonzero, then $T_{K,n}\to \infty$ in probability and the test is consistent. It has only null-level asymptotic power against means orthogonal to the first $K$ principal component functions. Under ${\mu }_n=n^{-1/2}\delta$, the limit is noncentral chi-squared with noncentrality

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.