= FPCA mean test
{c}
{title2=$T_{n,K}$}
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 $(\lambda_j,\phi_j)$ 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 is
$$
T_{n,K}=n\sum_{j=1}^K
\frac{\langle\overline X,\widehat\phi_j\rangle^2}{\widehat\lambda_j}.
$$
Under the <null hypothesis> $\mathbb EX_i=0$, the <central limit theorem> gives joint convergence of $\sqrt n\langle\overline X,\phi_j\rangle/\sqrt{\lambda_j}$ to $K$ <independent> standard <normal random variables>. <Statistical consistency> and <Slutsky's theorem> justify replacing the population <eigenvectors> and <eigenvalues> by their estimates. Consequently $T_{n,K}\xrightarrow{d}\chi_K^2$. The fixed-dimensional projected two-sample version has the same calibration.<Estimation of the Mean of Functional Time Series and a Two-Sample Problem by Lajos Horváth and others>[Section 3, Theorem 5, equations (3.13)–(3.15)]{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.
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