  Consider a scenario where we have access to train data with both covariates
and outcomes while test data only contains covariates. In this scenario, our
primary aim is to predict the missing outcomes of the test data. With this
objective in mind, we train parametric regression models under a covariate
shift, where covariate distributions are different between the train and test
data. For this problem, existing studies have proposed covariate shift
adaptation via importance weighting using the density ratio. This approach
averages the train data losses, each weighted by an estimated ratio of the
covariate densities between the train and test data, to approximate the
test-data risk. Although it allows us to obtain a test-data risk minimizer, its
performance heavily relies on the accuracy of the density ratio estimation.
Moreover, even if the density ratio can be consistently estimated, the
estimation errors of the density ratio also yield bias in the estimators of the
regression model's parameters of interest. To mitigate these challenges, we
introduce a doubly robust estimator for covariate shift adaptation via
importance weighting, which incorporates an additional estimator for the
regression function. Leveraging double machine learning techniques, our
estimator reduces the bias arising from the density ratio estimation errors. We
demonstrate the asymptotic distribution of the regression parameter estimator.
Notably, our estimator remains consistent if either the density ratio estimator
or the regression function is consistent, showcasing its robustness against
potential errors in density ratio estimation. Finally, we confirm the soundness
of our proposed method via simulation studies.
