Nonparametric Goodness-of-fit Testing under Covariate Shift

We study density ratio estimation and importance-weighted regression under target shift with continuous outputs. Under target shift, the conditional distribution of the inputs given the outputs remains invariant across the training and test distributions, while the output marginal distribution may change. Although this problem has been extensively studied for discrete outputs, the continuous setting is substantially less understood: the importance weights are determined by an unknown density ratio function, for which existing estimation methods lack explicit finite-sample convergence rates. We propose a spectral regularization method in a reproducing kernel Hilbert space (RKHS) for estimating the continuous density ratio from labeled training samples and unlabeled test inputs. Under a source condition with regularity parameter $ι>0$, we establish high-probability finite-sample guarantees and show that the estimator achieves the capacity-independent minimax-optimal RKHS-norm rate $O(n_η^{-ι/(2ι+2)})$. We then incorporate the estimated density ratio into importance-weighted regression and characterize the propagation of density-ratio estimation error to the final predictor. When sufficiently many samples are available for density ratio estimation, the resulting regression estimator attains the minimax-optimal rates of standard kernel regression. These results establish a finite-sample theory for continuous density ratio estimation and importance-weighted learning under target shift.

Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, $φ$-mixing, strong mixing, and $\mathcal{C}$-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.

Reweighting source samples to match a target covariate distribution is a standard response to distribution shift when generalizing evidence from one population to another. This strategy is well suited to deterministic, learnable covariate discrepancies, but can be insufficient when source--target population differences also contain changes beyond covariate shift or when estimation of the density-ratio weights is unstable. To address this challenge, we introduce a new model that allows non-systematic changes between two population laws after systematic shifts are accounted for. Such residual shift is modeled as random perturbations to the probability space that cannot be represented in a learnable way. In this way, we separate systematic shifts, treated as bias and corrected by reweighting, from residual random perturbations, treated as distributional uncertainty and handled through dataset pooling. Under pure random perturbations, this principle yields Augmented Inverse Distance Weighting (AIDW), which uses regression augmentation and variance-optimal dataset-level pooling. For mixed shifts, we develop Augmented Inverse Hybrid Weighting (AIHW), which interpolates between AIDW and standard augmented importance weighting. Both methods trade off sampling uncertainty and distributional uncertainty via a \emph{distributional distance} that describes the strength of random perturbations. We establish asymptotic properties of the methods, together with plug-in guidance for choosing tuning parameters and model diagnostic tools. Experiments on three real-world multi-site datasets demonstrate consistent reductions in mean-squared error compared with standard weighting baselines, along with substantially improved empirical coverage in settings where covariate-shift adjustment alone undercovers, showing the robustness of the proposed methods across diverse distribution shift scenarios.