The projected covariance measure for assumption-lean variable significance testing
提出一种投影协方差度量方法，用于在给定协变量Z时检验变量X对响应Y的条件均值独立性，通过灵活使用非参数或机器学习方法实现稳健的误差控制和高效能。
Testing the significance of a variable or group of variables X for predicting a response Y, given additional covariates Z, is a ubiquitous task in statistics. A simple but common approach is to specify a linear model, and then test whether the regression coefficient for X is nonzero. However, when the model is misspecified, the test may have poor power, for example, when X is involved in complex interactions, or lead to many false rejections. In this work, we study the problem of testing the model-free null of conditional mean independence, that is, that the conditional mean of Y given X and Z does not depend on X. We propose a simple and general framework that can leverage flexible nonparametric or machine learning methods, such as additive models or random forests, to yield both robust error control and high power. The procedure involves using these methods to perform regressions, first to estimate a form of projection of Y on X and Z using one-half of the data, and then to estimate the expected conditional covariance between this projection and Y on the remaining half of the data. While the approach is general, we show that a version of our procedure using spline regression achieves what we show is the minimax optimal rate in this nonparametric testing problem. Numerical experiments demonstrate the effectiveness of our approach both in terms of maintaining Type I error control, and power, compared to several existing approaches.