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Test and Measure for Partial Mean Dependence Based on Deep Neural Networks

Leheng CaiXu GuoWei Zhong
Dec 2022
摘要
It is of great importance to investigate the significance of a subset ofcovariates W for the response Y given covariates Z in regression modeling. Tothis end, we propose a new significance test for the partial mean independenceproblem based on deep neural networks and data splitting. The test statisticconverges to the standard chi-squared distribution under the null hypothesiswhile it converges to a normal distribution under the alternative hypothesis.We also suggest a powerful ensemble algorithm based on multiple data splittingto enhance the testing power. If the null hypothesis is rejected, we propose anew partial Generalized Measure of Correlation (pGMC) to measure the partialmean dependence of Y given W after controlling for the nonlinear effect of Z,which is an interesting extension of the GMC proposed by Zheng et al. (2012).We present the appealing theoretical properties of the pGMC and establish theasymptotic normality of its estimator with the optimal root-N converge rate.Furthermore, the valid confidence interval for the pGMC is also derived. As animportant special case when there is no conditional covariates Z, we alsoconsider a new test of overall significance of covariates for the response in amodel-free setting. We also introduce new estimator of GMC and derive itsasymptotic normality. Numerical studies and real data analysis are alsoconducted to compare with existing approaches and to illustrate the validityand flexibility of our proposed procedures.
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