  In this paper, we consider detecting and estimating breaks in heterogeneous
mean functions of high-dimensional functional time series which are allowed to
be cross-sectionally correlated and temporally dependent. A new test statistic
combining the functional CUSUM statistic and power enhancement component is
proposed with asymptotic null distribution theory comparable to the
conventional CUSUM theory derived for a single functional time series. In
particular, the extra power enhancement component enlarges the region where the
proposed test has power, and results in stable power performance when breaks
are sparse in the alternative hypothesis. Furthermore, we impose a latent group
structure on the subjects with heterogeneous break points and introduce an
easy-to-implement clustering algorithm with an information criterion to
consistently estimate the unknown group number and membership. The estimated
group structure can subsequently improve the convergence property of the
post-clustering break point estimate. Monte-Carlo simulation studies and
empirical applications show that the proposed estimation and testing techniques
have satisfactory performance in finite samples.
