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 $\ell_{\infty }$-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 $L_{p}$-norm (with $p\in [1,\infty )$) risks. By employing recent results in $k$-monotone estimation minimax lower bound theory, we show that these convergence rates are optimal.