Functional mean test

A functional mean test tests whether the mean of a Hilbert-space-valued random variable is zero. The squared-norm statistic $n\Vert \overline{X}{\Vert }^2$ has a weighted chi-squared limit under the null, while an FPCA test standardizes and truncates the coordinates.

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.