  In this paper, we generalize the metric-based permutation test for the
equality of covariance operators proposed by Pigoli et al. (2014) to the case
of multiple samples of functional data. To this end, the non-parametric
combination methodology of Pesarin and Salmaso (2010) is used to combine all
the pairwise comparisons between samples into a global test. Different
combining functions and permutation strategies are reviewed and analyzed in
detail. The resulting test allows to make inference on the equality of the
covariance operators of multiple groups and, if there is evidence to reject the
null hypothesis, to identify the pairs of groups having different covariances.
It is shown that, for some combining functions, step-down adjusting procedures
are available to control for the multiple testing problem in this setting. The
empirical power of this new test is then explored via simulations and compared
with those of existing alternative approaches in different scenarios. Finally,
the proposed methodology is applied to data from wheel running activity
experiments, that used selective breeding to study the evolution of locomotor
behavior in mice.
