We introduce a new test for conditional independence which is based on what we call the weighted generalised covariance measure (WGCM). It is an extension of the recently introduced generalised covariance measure (GCM). To test the null hypothesis of X and Y being conditionally independent given Z, our test statistic is a weighted form of the sample covariance between the residuals of nonlinearly regressing X and Y on Z. We propose different variants of the test for both univariate and multivariate X and Y. We give conditions under which the tests yield the correct type I error rate. Finally, we compare our novel tests to the original GCM using simulation and on real data sets. Typically, our tests have power against a wider class of alternatives compared to the GCM. This comes at the cost of having less power against alternatives for which the GCM already works well.

翻译：我们根据我们所称的加权通用共变措施(WGCM)引入了一个新的有条件独立测试。这是最近引入的通用共变措施(GCM)的延伸。为了测试X和Y在有条件上独立的无效假设(Z),我们的测试统计是非线性回退X和Y在Z上的样本共变的加权形式。我们对单向和多变量X和Y提出了不同的测试变量。我们给出了测试得出正确的I型误差率的条件。最后,我们用模拟和真实数据集将我们的新测试与原GCM进行对比。一般来说,我们的测试对比GCM更广泛的替代品类别具有威力。这样做的代价是,对GCM已经运转良好的替代品的威力较低。