Testing conditional mean independence for functional data

We propose a new nonparametric conditional mean independence test for a response variable $Y$ and a predictor variable $X$ where either or both can be function-valued. Our test is built on a new metric, the so-called functional martingale difference divergence, which fully characterizes the conditional mean dependence of $Y$ given $X$ and extends the martingale difference divergence proposed by Shao & Zhang (2014). We define an unbiased estimator of functional martingale difference divergence by using a $\mathcal{U}$-centring approach, and we obtain its limiting null distribution under mild assumptions. Since the limiting null distribution is not pivotal, we use the wild bootstrap method to estimate the critical value and show the consistency of the bootstrap test. Our test can detect the local alternative which approaches the null at the rate of $n^{-1/2}$ with a nontrivial power, where $n$ is the sample size. Unlike the three tests developed by Kokoszka et al. (2008), Lei (2014) and Patilea et al. (2016), our test does not require a finite-dimensional projection or assume a linear model, and it does not involve any tuning parameters. Promising finite-sample performance is demonstrated via simulations, and a real-data illustration is used to compare our test with existing ones.