Semiparametric Shape-restricted Estimators for Nonparametric Regression

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.