  Imputation and propensity score weighting are two popular techniques for
handling missing data. We address these problems using the regularized
M-estimation techniques in the reproducing kernel Hilbert space. Specifically,
we first use the kernel ridge regression to develop imputation for handling
item nonresponse. While 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 a novel
application of the maximum entropy method for the density ratio function
estimation. 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 the air pollution data
measured in Beijing, China.
