Distribution shifts are a serious concern in modern statistical learning as
they can systematically change the properties of the data away from the truth.
We focus on Wasserstein distribution shifts, where every data point may undergo
a slight perturbation, as opposed to the Huber contamination model where a
fraction of observations are outliers. We formulate and study shifts beyond
independent perturbations, exploring Joint Distribution Shifts, where the
per-observation perturbations can be coordinated. We analyze several important
statistical problems, including location estimation, linear regression, and
non-parametric density estimation. Under a squared loss for mean estimation and
prediction error in linear regression, we find the exact minimax risk, a least
favorable perturbation, and show that the sample mean and least squares
estimators are respectively optimal. This holds for both independent and joint
shifts, but the least favorable perturbations and minimax risks differ. For
other problems, we provide nearly optimal estimators and precise finite-sample
bounds. We also introduce several tools for bounding the minimax risk under
distribution shift, such as a smoothing technique for location families, and
generalizations of classical tools including least favorable sequences of
priors, the modulus of continuity, Le Cam's, Fano's, and Assouad's methods.

分布变化是现代统计学习中的一个严重问题，我们研究了 Wasserstein 分布变化以及联合分布变化，分析了一些重要的统计问题，包括位置估计、线性回归和非参数密度估计。对于均值估计和线性回归的预测误差，我们找到了最小二乘风险和最不利扰动，并证明了样本均值和最小二乘估计量分别是最优的。对于其他问题，我们提供了几乎最优的估计器和精确的有限样本界限。我们还引入了几种用于边界化分布变化的最小风险的工具，如平滑技术、最不利优先级序列的推广以及 Le Cam、Fano 和 Assouad 方法的泛化。