Difference-based methods have been attracting increasing attention in nonparametric regression, in particular for estimating the residual variance.To implement the estimation, one needs to choose an appropriate difference sequence, mainly between {\em the optimal difference sequence} and {\em the ordinary difference sequence}. The difference sequence selection is a fundamental problem in nonparametric regression, and it remains a controversial issue for over three decades. In this paper, we propose to tackle this challenging issue from a very unique perspective, namely by introducing a new difference sequence called {\em the optimal-$k$ difference sequence}. The new difference sequence not only provides a better balance between the bias-variance trade-off, but also dramatically enlarges the existing family of difference sequences that includes the optimal and ordinary difference sequences as two important special cases. We further demonstrate, by both theoretical and numerical studies, that the optimal-$k$ difference sequence has been pushing the boundaries of our knowledge in difference-based methods in nonparametric regression, and it always performs the best in practical situations.

翻译：基于差异的方法在非参数回归中引起了越来越多的注意,特别是用于估计剩余差异。为了执行这一估算,人们需要选择一个适当的差异序列,主要在最大差异序列和普通差异序列之间。差异序列的选择是非参数回归中的一个基本问题,30多年来它一直是一个有争议的问题。在本文件中,我们提议从非常独特的角度处理这个具有挑战性的问题,即采用称为最佳-美元差异序列的新差异序列。新的差异序列不仅在偏差交易之间提供了更好的平衡,而且还极大地扩大了现有差异序列的类别,其中包括最佳和普通差异序列,作为两个重要的特殊情况。我们通过理论和数字研究进一步表明,最佳-美元差异序列一直在推动我们在非参数回归中基于差异方法的知识的界限,而且总是在实际情况下实现最佳的。