Minimax Lower Bound and Optimal Estimation of Convex Functions in the Sup-Norm

Abstract

Estimation of convex functions finds broad applications in science and engineering; however, the convex shape constraint complicates the asymptotic performance analysis of such estimators. This technical note is devoted to the minimax optimal estimation of univariate convex functions in a given Hölder class. Particularly, a minimax lower bound in the supremum norm (or simply sup-norm) is established by constructing a novel family of piecewise quadratic convex functions in the Hölder class. This result, along with a recent result on the minimax upper bound, gives rise to the optimal rate of convergence for the minimax sup-norm risk of convex functions with the Hölder order between one and two. The present technical note provides the first rigorous justification of the optimal minimax risk for convex estimation on the entire interval of interest in the sup-norm.