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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

This paper addresses the problem of estimating a convex regression functionunder both the sup-norm risk and the pointwise risk using B-splines. Thepresence of the convex constraint complicates various issues in asymptoticanalysis, particularly uniform convergence analysis. To overcome thisdifficulty, we establish the uniform Lipschitz property of optimal splinecoefficients in the $\ell_\infty$-norm by exploiting piecewise linear andpolyhedral theory. Based upon this property, it is shown that this estimatorattains optimal rates of convergence on the entire interval of interest overthe H\"older class under both the risks. In addition, adaptive estimates areconstructed under both the sup-norm risk and the pointwise risk when theexponent of the H\"older class is between one and two. These estimates achievea maximal risk within a constant factor of the minimax risk over the H\"olderclass.

Uniform Convergence and Rate Adaptive Estimation of a Convex Function