Projection-averaging-based cumulative covariance and its use in goodness-of-fit testing for single-index models

Abstract

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.