From Isotonic to Lipschitz Regression: A New Interpolative Perspective on Shape-restricted Estimation

TL;DR

This paper introduces a unified framework connecting smoothness-based and shape-restricted estimation, proposing estimators with strong theoretical guarantees and broad applicability in multivariate and higher-order contexts.

Contribution

It generalizes the decomposition of Lipschitz functions into monotonic and linear parts, extending to higher-order and multivariate cases, with new estimators and theoretical analysis.

Findings

Proposed estimators have convergence guarantees under heteroscedastic and heavy-tailed errors.

The framework demonstrates strong approximation capabilities and numerical performance.

Establishes a unified perspective bridging different nonparametric estimation paradigms.

Abstract

This manuscript bridges nonparametric smoothness-based and shape-restricted estimation, which may appear as two disjoint paradigms in the field. The proposed approach is motivated by a conceptually simple observation: every Lipschitz function is a sum of a monotonic and a linear function. This principle is further generalized to the higher-order monotonicity and multivariate settings. A family of estimators is proposed based on a sample-splitting procedure, inheriting 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 to the unknown ``complexity" of the true regression function. The generality of the proposed decomposition framework is demonstrated through new approximation results and numerical studies.