标题： 基于核的回归在凸风险最小化中的相容性和稳健性 显示英文标题

标题： Consistency and robustness of kernel-based regression in convex risk minimization

摘要： 我们研究了一大类现代核回归（KBR）方法的统计性质。 这些核方法在过去十年中被开发出来，并受到无限维希尔伯特空间中凸风险最小化的启发。 一个典型的例子是支持向量回归。 我们首先描述了KBR方法的损失函数 $L$ 与响应变量尾部之间的关系。 然后我们建立了KBR的 $L$-风险一致性，这为“学习”这一说法提供了数学依据。 接着我们考虑了这类核方法的稳健性。 特别是，我们的结果使我们能够选择损失函数和核函数，以获得计算可行且一致的KBR方法，这些方法具有有界的影响力函数。 此外，我们还发展了偏差和敏感曲线（即影响函数的有限样本版本）的界，并讨论了KBR与经典 $M$ 估计量之间的关系。 显示英文摘要

摘要： We investigate statistical properties for a broad class of modern kernel-based regression (KBR) methods. These kernel methods were developed during the last decade and are inspired by convex risk minimization in infinite-dimensional Hilbert spaces. One leading example is support vector regression. We first describe the relationship between the loss function $L$ of the KBR method and the tail of the response variable. We then establish the $L$-risk consistency for KBR which gives the mathematical justification for the statement that these methods are able to ``learn''. Then we consider robustness properties of such kernel methods. In particular, our results allow us to choose the loss function and the kernel to obtain computationally tractable and consistent KBR methods that have bounded influence functions. Furthermore, bounds for the bias and for the sensitivity curve, which is a finite sample version of the influence function, are developed, and the relationship between KBR and classical $M$ estimators is discussed.