Composite quantile regression has been used to obtain robust estimators of regression coefficients in linear models with good statistical efficiency. By revealing an intrinsic link between the composite quantile regression loss function and the Wasserstein distance from the residuals to the set of quantiles, we establish a generalization of the composite quantile regression to the multiple-output settings. Theoretical convergence rates of the proposed estimator are derived both under the setting where the additive error possesses only a finite $\ell$-th moment (for $\ell > 2$) and where it exhibits a sub-Weibull tail. In doing so, we develop novel techniques for analyzing the M-estimation problem that involves Wasserstein-distance in the loss. Numerical studies confirm the practical effectiveness of our proposed procedure.

翻译：复合分位数回归已被用于在线性模型中获取具有良好统计效率的回归系数稳健估计量。通过揭示复合分位数回归损失函数与残差到分位数集合的Wasserstein距离之间的内在联系，我们建立了复合分位数回归到多输出场景的推广。在加性误差仅具有有限ℓ阶矩（ℓ>2）以及具有次威布尔尾部这两种设定下，我们推导了所提估计量的理论收敛速度。在此过程中，我们发展了分析涉及损失函数中Wasserstein距离的M估计问题的新技术。数值研究证实了我们所提方法的实际有效性。