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An FPCA mean test uses functional principal component analysis to test a zero expectation in a Hilbert space. For independent and identically distributed observations $X_i$ with finite second moment, let $({\lambda }_j,{\varphi }_j)$ be the eigenvalues and orthonormal eigenvectors of the covariance operator. Choose a fixed positive integer $K$ and, as a sufficient separation assumption, require ${\lambda }_1>\cdots >{\lambda }_K>{\lambda }_{K+1}$ and ${\lambda }_K>0$. With consistent estimators of these eigenvalues and eigenvectors, the statistic isUnder the null hypothesis $\mathbb{E}{X}_i=0$, the central limit theorem gives joint convergence of $\sqrt{n}⟨\overline{X},{\varphi }_j⟩/\sqrt{{\lambda }_j}$ to $K$ independent standard normal random variables. Statistical consistency and Slutsky's theorem justify replacing the population eigenvectors and eigenvalues by their estimates. Consequently $T_{n,K}\stackrel{d}{\to }{\chi }_K^2$. The fixed-dimensional projected two-sample version has the same calibration.^[ref].

The positivity condition prevents division by zero; the spectral gap makes the estimated directions stable. Repeated eigenvalues can be handled through an entire retained eigenspace, provided its boundary remains separated. A choice $K=K_n$ growing with $n$ needs additional approximation theory. A fixed $K$ can miss a nonzero expectation lying entirely in the omitted orthogonal directions, so the test's simplicity comes with a choice about which deviations matter.

A functional mean test asks whether the expectation $\mu$ of a Hilbert-space-valued random variable is zero. Suppose $X_1,\dots ,X_n$ are independent and identically distributed in a Hilbert space with a countable dense subset $H$, with $\mathbb{E}\Vert X_1{\Vert }^2<\infty$. Write $C$ for their covariance operator and $\overline{X}=n^{-1}{\sum }_i{X}_i$. Under the null hypothesis $\mu =0$, the Hilbert-space central limit theorem giveswhere $G$ is a centered Gaussian random element in $H$, the ${\lambda }_j$ are the eigenvalues of $C$, and the $Z_j$ are independent standard normal random variables. The second assertion follows by the continuous mapping theorem. Since ${\sum }_j{\lambda }_j=\mathbb{E}\Vert X_1-\mu {\Vert }^2<\infty$, the nonnegative series is finite almost surely. Limits given by Gaussian random elements for functional sample means and the corresponding weighted squared-norm laws also extend to suitable dependent observations.^[ref]

The eigenvalues determine the calibration: the limit is generally a weighted sum of random variables with a chi-squared distribution, rather than a single ordinary chi-squared distribution. The squared norm measures deviations across all covariance operator directions. An FPCA mean test instead retains finitely many eigenvectors and divides their squared coordinates by their eigenvalues, producing a different statistic with a simpler limiting calibration.

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.