Estimating the conditional mean function that relates predictive covariates to a response variable of interest is a fundamental task in statistics. In this paper, we propose some general nonparametric regression approaches that are widely applicable under very mild conditions. The method decomposes a function with a Lipschitz continuous $k$-th derivative into a sum of a $(k-1)$-monotone function and a parametric component. 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 rate, and exhibits a locally adaptive rate when the true regression function is "simple". Finally, a series of numerical studies are presented, confirming these theoretical properties.

翻译：估计连接预测协变量与感兴趣响应变量的条件均值函数是统计学中的基本任务。在本文中，我们提出了一些在非常温和条件下广泛适用的一般非参数回归方法。该方法将具有Lipschitz连续k阶导数的函数分解为一个(k-1)单调函数和一个参数分量之和。我们实施成熟的形状约束估计程序（如保序回归）来处理真实回归函数的“非参数”分量，并将其与简单的样本拆分程序结合以估计参数分量。所得估计器继承了形状约束回归估计器的若干优良性质。值得注意的是，它（在实际中）无需调整参数，以极小化最大速率收敛，并且在真实回归函数“简单”时展现出局部自适应速率。最后，通过一系列数值研究验证了这些理论性质。