Convex regression via penalized splines: A complementarity approach

Estimation of a convex function is an important shape restricted nonparametric inference problem with broad applications. In this paper, penalized splines (or simply P-splines) are exploited for convex estimation. The paper is devoted to developing an asymptotic theory of a class of P-spline convex estimators using complementarity techniques and asymptotic statistics. Due to the convex constraints, the optimality conditions of P-splines are characterized by nons-mooth complementarity conditions. A critical uniform Lipschitz property is established for optimal spline coefficients. This property yields boundary consistency and uniform stochastic boundedness. Using this property, the P-spline estimator is approximated by a two-step estimator based on the corresponding least squares estimator, and its asymptotic behaviors are obtained using asymptotic statistic techniques.