A novel metric, called kernel-based conditional mean dependence (KCMD), is proposed to measure and test the departure from conditional mean independence between a response variable Y and a predictor variable X, based on the reproducing kernel embedding and the Hilbert-Schmidt norm of a tensor operator. The KCMD has several appealing merits. It equals zero if and only if the conditional mean of Y given X is independent of X, i.e. $E(Y|X)=E(Y)$ almost surely, provided that the employed kernel is characteristic; it can be used to detect all kinds of conditional mean dependence with an appropriate choice of kernel; it has a simple expectation form and allows an unbiased empirical estimator. A class of test statistics based on the estimated KCMD is constructed, and a wild bootstrap test procedure to the conditional mean independence is presented. The limit distributions of the test statistics and the bootstrapped statistics under null hypothesis, fixed alternative hypothesis and local alternative hypothesis are given respectively, and a data-driven procedure to choose a suitable kernel is suggested. Simulation studies indicate that the tests based on the KCMD have close powers to the tests based on martingale difference divergence in monotone dependence, but excel in the cases of nonlinear relationships or the moment restriction on X is violated. Two real data examples are presented for the illustration of the proposed method.

提出了一种基于度量核嵌入和希尔伯特-施密特范数的新度量标准，即基于核的条件均值依赖性（KCMD），用于测量和测试响应变量Y和预测变量X之间条件均值独立性的偏离。张量运算符。KCMD具有几个吸引人的优点。当且仅当给定X的Y的条件均值独立于X时，它等于零，即$E（ÿ|X）=E（ÿ）$只要所使用的内核具有特征，就可以肯定地确定；可以通过选择适当的内核来检测各种条件均值依赖性。它具有简单的期望形式，并允许无偏的经验估计量。构造了基于估计的KCMD的一类测试统计量，并提出了一种对条件均值独立性的野生自举测试程序。分别给出了零假设，固定替代假设和局部替代假设下的检验统计量和自举统计量的极限分布，并提出了一种数据驱动的方法来选择合适的核。仿真研究表明，基于KCMD的测试与基于mar差在单调依赖性上的差异发散的测试具有接近的功效，X被违反。给出了两个真实的数据示例来说明所提出的方法。