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. The theoretical analysis provides convergence guarantees of the estimator under heteroscedastic and heavy-tailed errors, as well as adaptivity properties to the unknown 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 of the proposed estimation framework.

翻译：本文旨在弥合非参数回归中两个看似分离的范式：基于平滑性假设的估计与形状约束估计。所提出的方法源于一个概念上简单的观察：每个Lipschitz函数都是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。基于样本分割程序，我们提出了一族估计量，这些估计量继承了形状约束估计量在方法论、理论及计算方面的优良特性。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未知复杂度的自适应特性。通过新的逼近结果证明了所提出分解框架的普适性，大量数值研究验证了该估计框架的理论性质。