On Univariate Convex Regression

TL;DR

This paper analyzes the local convergence rates and limiting distributions of the univariate convex regression least squares estimator, especially at points where derivatives vanish or the function is affine, and proposes an estimator for the argmin.

Contribution

It provides new theoretical results on the convergence rates, limiting distributions, and boundary behavior of the convex LSE, including an estimator for the argmin.

Findings

Derived the local convergence rates of the convex LSE.

Established the limiting distribution depending on derivatives of the invelope function.

Demonstrated the inconsistency and boundary issues of the LSE.

Abstract

We find the local rate of convergence of the least squares estimator (LSE) of a one dimensional convex regression function when (a) a certain number of derivatives vanish at the point of interest, and (b) the true regression function is locally affine. In each case we derive the limiting distribution of the LSE and its derivative. The pointwise limiting distributions depend on the second and third derivatives at 0 of the "invelope function" of the integral of a two-sided Brownian motion with polynomial drifts. We also investigate the inconsistency of the LSE and the unboundedness of its derivative at the boundary of the domain of the covariate space. An estimator of the argmin of the convex regression function is proposed and its asymptotic distribution is derived. Further, we present some new results on the characterization of the convex LSE that may be of independent interest.