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We develop a group of robust, nonparametric hypothesis tests which detectdifferences between the covariance operators of several populations offunctional data. These tests, called FKWC tests, are based on functional datadepth ranks. These tests work well even when the data is heavy tailed, which isshown both in simulation and theoretically. These tests offer several otherbenefits, they have a simple distribution under the null hypothesis, they arecomputationally cheap and they possess transformation invariance properties. Weshow that under general alternative hypotheses these tests are consistent undermild, nonparametric assumptions. As a result of this work, we introduce a newfunctional depth function called L2-root depth which works well for thepurposes of detecting differences in magnitude between covariance kernels. Wepresent an analysis of the FKWC test using L2-root depth under localalternatives. In simulation, when the true covariance kernels have strictlypositive eigenvalues, we show that these tests have higher power than theircompetitors, while still maintaining their nominal size. We also provide amethods for computing sample size and performing multiple comparisons.

Robust nonparametric hypothesis tests for differences in the covariance structure of functional data