On Convex Least Squares Estimation when the Truth is Linear

TL;DR

This paper establishes the convergence rates and asymptotic distribution of the convex least squares estimator in regions where the true function is linear, enabling improved inference and testing procedures.

Contribution

It provides the first detailed analysis of the convex LSE's behavior in linear regions, including convergence rates, distributional limits, and adaptive properties at boundary points.

Findings

Convex LSE attains a $n^{-1/2}$ rate in linear regions.

Asymptotic distribution characterized by a modified invelope process.

Supports new tests for linearity against convex alternatives.

Abstract

We prove that the convex least squares estimator (LSE) attains a $n^{-1/2}$ pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent testing procedure on the linearity against a convex alternative. Moreover, we show that the convex LSE adapts to the optimal rate at the boundary points of the region where the truth is linear, up to a log-log factor. These conclusions are valid in the context of both density estimation and regression function estimation.