We propose an adjusted Wasserstein distributionally robust estimator -- based on a nonlinear transformation of the Wasserstein distributionally robust (WDRO) estimator in statistical learning. The classic WDRO estimator is asymptotically biased, while our adjusted WDRO estimator is asymptotically unbiased, resulting in a smaller asymptotic mean squared error. Meanwhile, the proposed adjusted WDRO has an out-of-sample performance guarantee. Further, under certain conditions, our proposed adjustment technique provides a general principle to de-bias asymptotically biased estimators. Specifically, we will investigate how the adjusted WDRO estimator is developed in the generalized linear model, including logistic regression, linear regression, and Poisson regression. Numerical experiments demonstrate the favorable practical performance of the adjusted estimator over the classic one.

翻译：我们提出了一种基于Wasserstein分布鲁棒（WDRO）估计量非线性变换的调整型Wasserstein分布鲁棒估计量——该估计量应用于统计学习。经典的WDRO估计量存在渐近偏差，而本文提出的调整型WDRO估计量具有渐近无偏性，从而获得更小的渐近均方误差。同时，该调整型WDRO估计量具备样本外性能保障。进一步，在特定条件下，本文提出的调整技术为渐近有偏估计量的去偏提供了通用原则。具体而言，我们将研究调整型WDRO估计量在广义线性模型（包括逻辑回归、线性回归和泊松回归）中的构建方法。数值实验表明，相较于经典估计量，调整型估计量在实际应用中具有更优的性能表现。