 Solution

Letand estimate it by the average of the two within-sample empirical covariance operators. Let $({\hat{\lambda }}_k,{\hat{\varphi }}_k)$ be its leading empirical eigenpairs and let ${\overline{X}}_n,{\overline{X}}_n^{\ast }$ be the sample means. Use the Two-sample FPCA mean statistic

Under $H_0$, the Hilbert-space central limit theorem giveswhere $G$ is a centered Gaussian random element with covariance $\overline{C}$. Distinct eigenvalues give consistent empirical eigenpairs, up to signs, and the standardized leading scores are independent standard normal variables. HenceRejecting above the $(1-\alpha )$ quantile gives an asymptotic level-$\alpha$ test.

Under a fixed alternative $\delta =\mu -{\mu }^{\ast }$,The test is consistent whenever one retained projection is nonzero. Alternatives orthogonal to the first $K$ eigenfunctions are invisible at fixed $K$. Under local alternatives $\delta =h/\sqrt{n}$, the limit is noncentral chi-squared with noncentrality $\frac{1}{2}{\sum }_{k\le K}⟨h,{\varphi }_k{⟩}^2/{\lambda }_k$.