In this paper, we generalize the metric-based permutation test for theequality of covariance operators proposed by Pigoli et al. (2014) to the caseof multiple samples of functional data. To this end, the non-parametriccombination methodology of Pesarin and Salmaso (2010) is used to combine allthe pairwise comparisons between samples into a global test. Differentcombining functions and permutation strategies are reviewed and analyzed indetail. The resulting test allows to make inference on the equality of thecovariance operators of multiple groups and, if there is evidence to reject thenull hypothesis, to identify the pairs of groups having different covariances.It is shown that, for some combining functions, step-down adjusting proceduresare available to control for the multiple testing problem in this setting. Theempirical power of this new test is then explored via simulations and comparedwith those of existing alternative approaches in different scenarios. Finally,the proposed methodology is applied to data from wheel running activityexperiments, that used selective breeding to study the evolution of locomotorbehavior in mice.
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