English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Estimation of convex functions finds broad applications in engineering andscience, while convex shape constraint gives rise to numerous challenges inasymptotic performance analysis. This paper is devoted to minimax optimalestimation of univariate convex functions from the H\"older class in theframework of shape constrained nonparametric estimation. Particularly, thepaper establishes the optimal rate of convergence in two steps for the minimaxsup-norm risk of convex functions with the H\"older order between one and two.In the first step, by applying information theoretical results on probabilitymeasure distance, we establish the minimax lower bound under the supreme normby constructing a novel family of piecewise quadratic convex functions in theH\"older class. In the second step, we develop a penalized convex splineestimator and establish the minimax upper bound under the supreme norm. Due tothe convex shape constraint, the optimality conditions of penalized convexsplines are characterized by nonsmooth complementarity conditions. Byexploiting complementarity methods, a critical uniform Lipschitz property ofoptimal spline coefficients in the infinity norm is established. This property,along with asymptotic estimation techniques, leads to uniform bounds for biasand stochastic errors on the entire interval of interest. This further yieldsthe optimal rate of convergence by choosing the suitable number of knots andpenalty value. The present paper provides the first rigorous justification ofthe optimal minimax risk for convex estimation under the supreme norm.

Minimax Optimal Estimation of Convex Functions in the Supreme Norm