Statistical inference using regularized M-estimation in the reproducing kernel Hilbert space for handling missing data

Hengfang Wang, Jae Kwang Kim

Abstract: Imputation is a popular technique for handling missing data. We address a nonparametric imputation using the regularized M-estimation techniques in the reproducing kernel Hilbert space. Specifically, we first use kernel ridge regression to develop imputation for handling item nonresponse. Although this nonparametric approach is potentially promising for imputation, its statistical properties are not investigated in the literature. Under some conditions on the order of the tuning parameter, we first establish the root-$n$ consistency of the kernel ridge regression imputation estimator and show that it achieves the lower bound of the semiparametric asymptotic variance. A nonparametric propensity score estimator using the reproducing kernel Hilbert space is also developed by the linear expression of the projection estimator. We show that the resulting propensity score estimator is asymptotically equivalent to the kernel ridge regression imputation estimator. Results from a limited simulation study are also presented to confirm our theory. The proposed method is applied to analyze air pollution data measured in Beijing, China.