A new similarity measure for covariate shift with applications to nonparametric regression

In the standard formulation of prediction or classification, future data (as represented by a test set) is assumed to be drawn from the same distribution as the training data. This assumption, while theoretically convenient, may fail to hold in many real-world scenarios. For instance, training data might be collected only from a sub-group within a broader population (such as in medical trials), or the environment might change over time as data are collected. Such scenarios result in a distribution mismatch between the training and test data. In this paper, we study an important case of such distribution mismatch--namely, the covariate shift problem (e.g., [21, 19]). Suppose that a statistician observes covariate-response pairs (X, Y), and wishes to build a prediction rule. In the problem of covariate shift, the distribution of the covariates X is allowed to change between the training and test data, while the posterior distribution of the responses (namely, Y X) remains fixed. Compared to the usual i.i.d.