Estimating the conditional mean function that relates predictive covariates to a response variable of interest is a fundamental task in economics and statistics. In this manuscript, we propose some general nonparametric regression approaches that are widely applicable based on a simple yet significant decomposition of nonparametric functions into a semiparametric model with shape-restricted components. For instance, we observe that every Lipschitz function can be expressed as a sum of a monotone function and a linear function. We implement well-established shape-restricted estimation procedures, such as isotonic regression, to handle the ``nonparametric" components of the true regression function and combine them with a simple sample-splitting procedure to estimate the parametric components. The resulting estimators inherit several favorable properties from the shape-restricted regression estimators. Notably, it is practically tuning parameter free, converges at the minimax optimal rate, and exhibits an adaptive rate when the true regression function is ``simple". We also confirm these theoretical properties and compare the practice performance with existing methods via a series of numerical studies.

翻译：本文旨在估计条件均值函数，该函数将预测协变量与感兴趣的响应变量联系起来，是经济学和统计学中的基本任务。本文基于非参数函数的一种简单而重要的分解方法，将其分解为具有形状约束成分的半参数模型，提出了一些具有广泛适用性的通用非参数回归方法。例如，我们观察到，每个Lipschitz函数都可以表示为单调函数与线性函数之和。我们采用已成熟的形状约束估计程序（如等温回归）来处理真实回归函数中的“非参数”成分，并将其与简单的样本分割程序相结合以估计参数成分。由此得到的估计量继承了形状约束回归估计量的若干优良性质。特别地，该估计量在实际应用中几乎无需调整参数，以极小化最优速率收敛，并且在真实回归函数较为“简单”时表现出自适应速率。我们通过一系列数值研究验证了这些理论性质，并将其实践性能与现有方法进行了比较。