A two stage k-monotone B-spline regression estimator: Uniform Lipschitz property and optimal convergence rate

Abstract

This paper considers k-monotone estimation and the related asymptotic performance analysis over a suitable Hölder class for general k. A novel two-stage k-monotone B-spline estimator is proposed: in the first stage, an unconstrained estimator with optimal asymptotic performance is considered; in the second stage, a k-monotone B-spline estimator is constructed (roughly) by projecting the unconstrained estimator onto a cone of k-monotone splines. To study the asymptotic performance of the second-stage estimator under the sup-norm and other risks, a critical uniform Lipschitz property for the k-monotone B-spline estimator is established under the ℓ∞-norm. This property uniformly bounds the Lipschitz constants associated with the mapping from a (weighted) first-stage input vector to the B-spline coefficients of the second-stage k-monotone estimator, independent of the sample size and the number of knots. This result is then exploited to analyze the second-stage estimator performance and develop convergence rates under the sup-norm, pointwise, and Lp-norm (with p∈[1,∞)) risks. By employing recent results in k-monotone estimation minimax lower bound theory, we show that these convergence rates are optimal.