We develop nonparametric regression methods for the case when the true regression function is not necessarily smooth. More specifically, our approach is using the fractional Laplacian and is designed to handle the case when the true regression function lies in an $L_2$-fractional Sobolev space with order $s\in (0,1)$. This function class is a Hilbert space lying between the space of square-integrable functions and the first-order Sobolev space consisting of differentiable functions. It contains fractional power functions, piecewise constant or polynomial functions and bump function as canonical examples. For the proposed approach, we prove upper bounds on the in-sample mean-squared estimation error of order $n^{-\frac{2s}{2s+d}}$, where $d$ is the dimension, $s$ is the aforementioned order parameter and $n$ is the number of observations. We also provide preliminary empirical results validating the practical performance of the developed estimators.

翻译：本文针对真实回归函数不一定光滑的情况，发展了非参数回归方法。具体而言，我们的方法基于分数阶拉普拉斯算子，旨在处理真实回归函数属于阶数$s\in(0,1)$的$L_2$分数阶Sobolev空间的情形。这一函数类是介于平方可积函数空间与由可微函数构成的一阶Sobolev空间之间的希尔伯特空间，其典型例子包括分数幂函数、分段常数或多项式函数以及隆起函数。对于所提方法，我们证明了样本内均方估计误差的上界为$n^{-\frac{2s}{2s+d}}$，其中$d$为维度，$s$为前述阶数参数，$n$为观测数量。我们还提供了初步的实证结果，验证了所开发估计量的实际性能。