A projection-averaging-based cumulative divergence to characterize the conditional mean independence is proposed. As a natural extension of Zhou et al. (2020), the new metric has several appealing features. It ranges from zero to one, and equals zero if and only if the conditional mean independence holds. It has an elegant closed-form expression that involves no tuning parameters, making it easy to implement. The sample estimator of new metric is n-consistent under the conditional mean independence and root-n-consistent otherwise. A goodness-of-fit test for single-index models based on the variant of the proposed metric is further introduced, which generalizes the projected-based test of Escanciano (2006) to a semiparametric regression setting that allows an unspecified link function. The proposed test is consistent against any global alternatives and can detect the local alternatives distinct from the null at the parametric rate of $O(n^{-1/2})$. The effectiveness of our proposals is demonstrated through simulation examples and a real application.

提出了一种基于投影平均的累积散度来表征条件平均独立性。作为 Zhou 等人的自然延伸。(2020)，新指标有几个吸引人的特点。它的范围从 0 到 1，并且当且仅当条件平均独立性成立时才等于 0。它有一个优雅的封闭式表达式，不涉及调整参数，使其易于实现。新指标的样本估计是ň条件均值独立性和根-下-consistent ñ- 否则保持一致。进一步介绍了基于所提出度量的变体的单指数模型的拟合优度检验，它将 Escanciano (2006) 的基于投影的检验推广到允许未指定链接函数的半参数回归设置。建议的测试与任何全局替代方案一致，并且可以以参数速率检测与零不同的局部替代方案$哦(n^{-1/2})$. 通过模拟示例和实际应用证明了我们建议的有效性。