  Covariate shift is a common transfer learning scenario where the marginal
distributions of input variables vary between source and target data while the
conditional distribution of the output variable remains consistent. The
existing notions describing differences between marginal distributions face
limitations in handling scenarios with unbounded support, particularly when the
target distribution has a heavier tail. To overcome these challenges, we
introduce a new concept called density ratio exponent to quantify the relative
decay rates of marginal distributions' tails under covariate shift.
Furthermore, we propose the local k-nearest neighbour regressor for transfer
learning, which adapts the number of nearest neighbours based on the marginal
likelihood of each test sample. From a theoretical perspective, convergence
rates with and without supervision information on the target domain are
established. Those rates indicate that our estimator achieves faster
convergence rates when the density ratio exponent satisfies certain conditions,
highlighting the benefits of using density estimation for determining different
numbers of nearest neighbours for each test sample. Our contributions enhance
the understanding and applicability of transfer learning under covariate shift,
especially in scenarios with unbounded support and heavy-tailed distributions.
