Title: On Univariate Convex Regression

Author(s): Promit Ghosal and Bodhisattva Sen
Issue: Volume 79 Series A Part 2 Year 2017
Pages: 215 -- 253
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.
AMS (2000) subject classification. Primary 62G08, 62G05; Secondary 62G20.
Keywords and phrases: Convex function estimation, Integral of Brownian motion, Invelope process, Least squares estimator, Shape constrained regression.