Distance of mean embedding for testing independence of functional data

We investigate independence testing for functional data, which maybe either univariate or multivariate. Broadly speaking, our approach involves first reducing the dimensionality of the functional data using basis expansion and then applying the distance of mean embedding- a flexible measure of independence. We enhance this method for pairwise independence by incorporating marginal aggregation, as well as asymmetric and symmetric aggregation measures, to improve test performance and adapt it to mutual independence testing. Our methods are compared with tests based on distance covariance and the Hilbert-Schmidt independence criterion. To evaluate their effectiveness, we present simulation studies and two real data examples using air pollution and chemometric data sets. The new testing procedures demonstrate favorable finite-sample properties, effectively controlling the type I error rate and exhibiting competitive power, making them viable alternatives to covariance-based tests.