Title: Conditional Mean and Quantile Dependence Testing in High Dimension,
Abstract: Motivated by applications in biological science, we propose a novel test to
assess the conditional mean dependence of a response variable on a large number
of covariates. Our procedure is built on the martingale difference divergence
recently proposed in Shao and Zhang (2014), and it is able to detect a certain
type of departure from the null hypothesis of conditional mean independence
without making any specific model assumptions. Theoretically, we establish the
asymptotic normality of the proposed test statistic under suitable assumption
on the eigenvalues of a Hermitian operator, which is constructed based on the
characteristic function of the covariates. These conditions can be simplified
under banded dependence structure on the covariates or Gaussian design. To
account for heterogeneity within the data, we further develop a testing
procedure for conditional quantile independence at a given quantile level and
provide an asymptotic justification. Empirically, our test of conditional mean
independence delivers comparable results to the competitor, which was
constructed under the linear model framework, when the underlying model is
linear. It significantly outperforms the competitor when the conditional mean
admits a nonlinear form.