Estimating Monotone Convex Functions via Sequential Shape Modification

We propose a sequential method to estimate monotone convex function that consists of: (i) monotone regression via solving a constrained least square problem and (ii) convexification of the monotone regression estimate via solving an associated constrained uniform approximation problem. We show that this method is faster than the constrained least squares (LS) method. The ratio of computation time increases as data size increases. Moreover, we show that, under an appropriate smoothness condition, the uniform convergence rate achieved by the proposed method is nearly comparable to the best achievable rate for a nonparametric estimate which ignores the shape constraint. Simulation studies show that our method is comparable to the constrained LS method in estimation error. We illustrate our method by analyzing ground water level data of wells in Korea.