We propose an L2-norm based global testing procedure
for the null hypothesis that multiple group mean functions are equal, for functional data
with complex dependence structure. Specifically, we consider the setting of functional
data with a multilevel structure of the form groups–clusters or
subjects–units, where the unit-level profiles are spatially correlated within the
cluster, and the cluster-level data are independent. Orthogonal series expansions are used
to approximate the group mean functions and the test statistic is estimated using the
basis coefficients. The asymptotic null distribution of the test statistic is developed,
under mild regularity conditions. To our knowledge this is the first work that studies
hypothesis testing, when data have such complex multilevel functional and spatial
structure. Two small-sample alternatives, including a novel block bootstrap for functional
data, are proposed, and their performance is examined in simulation studies. The paper
concludes with an illustration of a motivating experiment.