Uniform Convergence and Rate Adaptive Estimation of a Convex Function

TL;DR

This paper develops uniform convergence results and rate-adaptive estimators for convex functions using B-splines, overcoming challenges posed by convex constraints in asymptotic analysis.

Contribution

It establishes the uniform Lipschitz property of spline coefficients and constructs adaptive estimators that achieve near-minimax rates over H"older classes.

Findings

Estimator attains optimal convergence rates under sup-norm and pointwise risks.

Adaptive estimates are effective for H"older exponents between one and two.

Uniform Lipschitz property of spline coefficients is proven for convex regression.

Abstract

This paper addresses the problem of estimating a convex regression function under both the sup-norm risk and the pointwise risk using B-splines. The presence of the convex constraint complicates various issues in asymptotic analysis, particularly uniform convergence analysis. To overcome this difficulty, we establish the uniform Lipschitz property of optimal spline coefficients in the $\ell_\infty$-norm by exploiting piecewise linear and polyhedral theory. Based upon this property, it is shown that this estimator attains optimal rates of convergence on the entire interval of interest over the H\"older class under both the risks. In addition, adaptive estimates are constructed under both the sup-norm risk and the pointwise risk when the exponent of the H\"older class is between one and two. These estimates achieve a maximal risk within a constant factor of the minimax risk over the H\"older class.