A novel regression method is introduced and studied. The procedure weights squared residuals based on their magnitude. Unlike the classic least squares which treats every squared residual equally important, the new procedure exponentially down-weights squared-residuals that lie far away from the cloud of all residuals and assigns a constant weight (one) to squared-residuals that lie close to the center of the squared-residual cloud. The new procedure can keep a good balance between robustness and efficiency, it possesses the highest breakdown point robustness for any regression equivariant procedure, much more robust than the classic least squares, yet much more efficient than the benchmark of robust method, the least trimmed squares (LTS) of Rousseeuw (1984). With a smooth weight function, the new procedure could be computed very fast by the first-order (first-derivative) method and the second-order (second-derivative) method. Assertions and other theoretical findings are verified in simulated and real data examples.

翻译：本文提出并研究了一种新颖的回归方法。该方法依据平方残差的幅值进行加权。与经典最小二乘法将所有平方残差视为同等重要不同，新方法对远离残差云团的平方残差进行指数级降权，而对靠近平方残差云团中心的平方残差赋予恒定权重（权重为1）。该方法能在鲁棒性和效率之间保持良好平衡：对于任何回归等变方法，它均拥有最高的崩溃点鲁棒性，其鲁棒性远超经典最小二乘法，且效率显著高于鲁棒方法基准——Rousseeuw（1984）提出的最小修剪平方（LTS）法。凭借平滑的权重函数，该方法可通过一阶（一阶导数）和二阶（二阶导数）方法实现快速计算。模拟与真实数据案例验证了本文的论断及其他理论发现。