Uniform Lipschitz Property of Nonnegative Derivative Constrained B-Splines and Applications to Shape Constrained Estimation

Abstract

Inspired by shape constrained estimation under general nonnegative derivative constraints, this paper considers the B-spline approximation of constrained functions and studies the asymptotic performance of the constrained B-spline estimator. By invoking a deep result in B-spline theory (known as de Boor's conjecture) first proved by A. Shardin as well as other new analytic techniques, we establish a critical uniform Lipschitz property of the B-spline estimator subject to arbitrary nonnegative derivative constraints under the $\ell_\infty$-norm with possibly non-equally spaced design points and knots. This property leads to important asymptotic analysis results of the B-spline estimator, e.g., the uniform convergence and consistency on the entire interval under consideration. The results developed in this paper not only recover the well-studied monotone and convex approximation and estimation as special cases, but also treat general nonnegative derivative constraints in a unified framework and open the door for the constrained B-spline approximation and estimation subject to a broader class of shape constraints.