From isotonic to Lipschitz regression: a new interpolative perspective on shape-restricted estimation

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.