Doubly Robust Covariate Shift Regression with Semi-nonparametric Nuisance Models
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Importance weighting is naturally used to adjust for covariate shift. However, this simple strategy is not robust to model misspecifcation or excessive estimation error. To handle this problem, we propose a doubly robust covariate shift regression method by introducing an imputation model for conditional mean of the response to augment the importance weighted estimating equation. With a novel semi-nonparametric construction on the nuisance importance weight and imputation model, our method is more robust to excessive fitting error compared to the existing nonparametric or machine learning approaches and less likely to suffer from model misspecification than the parametric approach. To remove the overfitting bias of the nonparametric components under potential model misspecification, we construct specific calibrated moment estimating equations for the semi-nonparametric models. Theoretical properties of our estimator are studied. We show that our estimator attains the parametric rate when at least one nuisance model is correctly specified, estimation for the parametric part of the nuisance models achieves parametric rate and the nonparametric components satisfy the rate double robustness property. Simulation studies demonstrate that our method is more robust and efficient than parametric and fully nonparametric (machine learning) estimators under various configurations. We also examine the utility of our method through a real example about transfer learning of phenotyping algorithm for bipolar disorder. Finally, we discuss on possible ways to improve the (intrinsic) efficiency of our estimator and the potentiality of incorporating other nonparametric, high dimensional and machine learning models with our proposed framework.
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