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

TL;DR

This paper develops a novel nonparametric approach using regularized M-estimation in reproducing kernel Hilbert spaces for handling missing data, providing theoretical guarantees and practical applications.

Contribution

It introduces kernel ridge regression for imputation and a kernel-based propensity score estimator, both with proven statistical properties and asymptotic equivalence.

Findings

Kernel ridge regression imputation is root-n consistent.

Proposed propensity score estimator is asymptotically equivalent to the imputation estimator.

Simulation results support the theoretical properties.

Abstract

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.