Minimax Lower Bound and Optimal Estimation of Convex Functions in the Sup-Norm
Loading...
Links to Files
Author/Creator
Author/Creator ORCID
Date
2017-07
Type of Work
Department
Program
Citation of Original Publication
Lebair, Teresa M., Jinglai Shen, and Xiao Wang. “Minimax Lower Bound and Optimal Estimation of Convex Functions in the Sup-Norm.” IEEE Transactions on Automatic Control 62, no. 7 (July 2017): 3482–87. https://doi.org/10.1109/TAC.2016.2612543.
Rights
© 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
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.