This manuscript seeks to bridge two seemingly disjoint paradigms of nonparametric regression estimation based on smoothness assumptions and shape constraints. The proposed approach is motivated by a conceptually simple observation: Every Lipschitz function is a sum of monotonic and linear functions. This principle is further generalized to the higher-order monotonicity and multivariate covariates. A family of estimators is proposed based on a sample-splitting procedure, which inherits desirable methodological, theoretical, and computational properties of shape-restricted estimators. Our theoretical analysis provides convergence guarantees of the estimator under heteroscedastic and heavy-tailed errors, as well as adaptive properties to the complexity of the true regression function. The generality of the proposed decomposition framework is demonstrated through new approximation results, and extensive numerical studies validate the theoretical properties and empirical evidence for the practicalities of the proposed estimation framework.

翻译：本文旨在弥合基于平滑性假设与形状约束的两类非参数回归估计范式。该方法的提出源于一个概念上简洁的观察：所有Lipschitz函数均可表示为单调函数与线性函数之和。该原理被进一步推广至高阶单调性与多元协变量情形。基于样本分割策略，我们提出了一族估计量，其继承了形状约束估计量在方法学、理论及计算层面的优良特性。理论分析证明了该估计量在异方差与重尾误差下的收敛保证，以及对真实回归函数复杂度的自适应性质。通过新的逼近结果展示了所提分解框架的普适性，大量数值研究验证了理论性质，并为该估计框架的实际应用提供了实证依据。