Testing and measuring the conditional mean (in)dependence for functional data by martingale difference-angle divergence

Abstract

We proposed a new nonparametric method to test and measure conditional mean (in)dependence for functional data. This new metric has some appealing properties: it is nonnegative and equals to zero if and only if the conditional mean independence holds; it is invariant under linear transformations of the predictor; it does not require the moment condition for the predictor variable. Based on this measure, two test procedures for conditional mean independence are proposed for functional data. One uses a wild bootstrap while the other uses the limiting standard normal distribution. The tests are consistent and perform well in finite sample simulations. We further propose some requirements for a reasonable conditional mean dependence measure and demonstrate that our metric has those properties. A real data example is introduced to illustrate the application of the proposed method.