FPCA mean test

For covariance eigenpairs $({\lambda }_k,{\varphi }_k)$, the $K$-coordinate FPCA mean statistic is $n{\sum }_{k=1}^K⟨\overline{X},{\varphi }_k{⟩}^2/{\lambda }_k$. Under a zero-mean null and standard estimation conditions, its plug-in version converges to a chi-squared distribution with $K$ degrees of freedom.