论文标题
Lipschitz假设下的Nadaraya-Watson内核回归的上限
An Upper Bound of the Bias of Nadaraya-Watson Kernel Regression under Lipschitz Assumptions
论文作者
论文摘要
Nadaraya-Watson内核估计器由于其简单性而成为最受欢迎的非参数回归技术之一。罗森布拉特（Rosenblatt）于1969年对其渐近偏见进行了研究，并在许多相关文献中进行了报道。但是，Rosenblatt的分析仅适用于无穷小带宽。相比之下，我们在本文中提出了一个偏差的上限，该偏差具有有限的带宽。此外，与经典分析相反，我们允许回归函数的不连续的一阶导数，我们扩展了多维域的界限，并且在存在时包括回归函数的界限的知识，并且如果已知，则获得了更严格的界限。我们认为，这项工作在需要一些难以保证错误的领域具有潜在的应用
The Nadaraya-Watson kernel estimator is among the most popular nonparameteric regression technique thanks to its simplicity. Its asymptotic bias has been studied by Rosenblatt in 1969 and has been reported in a number of related literature. However, Rosenblatt's analysis is only valid for infinitesimal bandwidth. In contrast, we propose in this paper an upper bound of the bias which holds for finite bandwidths. Moreover, contrarily to the classic analysis we allow for discontinuous first order derivative of the regression function, we extend our bounds for multidimensional domains and we include the knowledge of the bound of the regression function when it exists and if it is known, to obtain a tighter bound. We believe that this work has potential applications in those fields where some hard guarantees on the error are needed
