Efficient estimation of smoothing spline with exact shape constraints

Smoothing spline is a popular method in non-parametric function estimation. For the analysis of data from real applications, specific shapes on the estimated function are often required to ensure the estimated function undeviating from the domain knowledge. In this work, we focus on constructing the exact shape constrained smoothing spline with efficient estimation. The 'exact' here is referred as to impose the shape constraint on an infinite set such as an interval in one-dimensional case. Thus the estimation becomes a so-called semi-infinite optimisation problem with an infinite number of constraints. The proposed method is able to establish a sufficient and necessary condition for transforming the exact shape constraints to a finite number of constraints, leading to efficient estimation of the shape constrained functions. The performance of the proposed methods is evaluated by both simulation and real case studies.