  We study covariate shift in the context of nonparametric regression. We
introduce a new measure of distribution mismatch between the source and target
distributions that is based on the integrated ratio of probabilities of balls
at a given radius. We use the scaling of this measure with respect to the
radius to characterize the minimax rate of estimation over a family of H\"older
continuous functions under covariate shift. In comparison to the recently
proposed notion of transfer exponent, this measure leads to a sharper rate of
convergence and is more fine-grained. We accompany our theory with concrete
instances of covariate shift that illustrate this sharp difference.
