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

We develop a group of robust, nonparametric hypothesis tests that detect differences between the covariance operators of several populations of functional data. These tests, called functional Kruskal–Wallis tests for covariance, or FKWC tests, are based on functional data depth ranks. FKWC tests work well even when the data are heavy-tailed, which is shown both in simulation and theory. FKWC tests offer several other benefits: they have a simple asymptotic distribution under the null hypothesis, they are computationally cheap, and they possess transformation-invariance properties. We show that under general alternative hypotheses, these tests are consistent under mild, nonparametric assumptions. As a result, we introduce a new functional depth function called $L^2$-root depth that works well for the purposes of detecting differences in magnitude between covariance kernels. We present an analysis of the FKWC test based on $L^2$-root depth under local alternatives. Through simulations, when the true covariance kernels have an infinite number of positive eigenvalues, we show that these tests have higher power than their competitors while maintaining their nominal size. We also provide a method for computing sample size and performing multiple comparisons.