 FPCA mean test

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.