 Solution

HereIt has the same eigenfunctions as $C_X$, but the $(K+1)$st eigenvalue is raised from ${\lambda }_{K+1}$ to ${\lambda }_{K+1}+b^2$. A test based on the first $K$ eigenfunctions of $C_X$ is blind to this alternative because $\delta =b{\varphi }_{K+1}$ is orthogonal to all of them. If the raised eigenvalue overtakes at least ${\lambda }_K$, then ${\varphi }_{K+1}$ enters the first $K$ eigenfunctions of $C^{\ast }$ and the mean shift acquires a retained nonzero coordinate. The $C^{\ast }$-based test can therefore detect alternatives that the original FPCA truncation misses.