2023年青年统计学家春季论坛｜钟威教授：偏均值相依性的假设检验和统计度量

It is of importance to investigate the significance of a subset of covariates W for the response Y given covariates Z in regression modeling. To this end, we propose a significance test for partial mean independence based on deep neural networks and data splitting. The test statistic converges to the standard chi-squared distribution under the null hypothesis while it converges to a normal distribution under the alternative hypothesis. We suggest a powerful ensemble algorithm based on multiple data splitting to enhance the testing power. If the null hypothesis is rejected, we propose a partial Generalized Measure of Correlation (pGMC) to measure the partial mean dependence of Y given W after controlling for the nonlinear effect of Z. We present the theoretical properties of the pGMC and establish the asymptotic normality of its estimator with the optimal root-N converge rate. Furthermore, the valid confidence interval for the pGMC is also derived. As an important special case when there is no conditional covariates Z, we consider a new test of overall significance of covariates for the response in a model-free setting. Numerical studies and real data analysis are conducted to compare with existing approaches and to illustrate the validity of our procedures.