FPCA mean test ($T_{n,K}$)

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.

Let $X_1,\dots ,X_n$ and $Y_1,\dots ,Y_n$ be two independent samples of independent and identically distributed Hilbert-space-valued random variables, with finite second moments and covariance operators $C_X,C_Y$. To test equality of their expectations, use the pooled covariance operator $C=(C_X+C_Y)/2$. This choice works even when the two population covariance operators differ: under the null hypothesis, the covariance operator of $\sqrt{n/2}(\overline{X}-\overline{Y})$ is exactly $C$.

Let $({\hat{\lambda }}_j,{\hat{\varphi }}_j)$ estimate the eigenvalues and orthonormal eigenvectors of $C$, for example from the average of the two within-sample empirical covariance operators. For fixed $K$, positive retained eigenvalues, a separated retained eigenspace and suitable statistical consistency of these estimates,The proof applies the Hilbert-space central limit theorem to the difference of sample means, then standardizes its first $K$ coordinates and uses Slutsky's theorem. A sufficient simple-eigenvalue condition is ${\lambda }_1>\cdots >{\lambda }_K>{\lambda }_{K+1}$ with ${\lambda }_K>0$.^[ref]

For unequal sample sizes $n,m$, the scaling becomes $nm/(n+m)$ and the relevant population covariance operator is $\{m{C}_X+n{C}_Y\}/(n+m)$. Thus a sample-size-weighted pooled estimate cannot automatically be reused for unequal sizes and unequal covariance operators: its weights must match the variance of the difference being tested.