Summary: Given noisy data, function estimation is considered when the unknown function is known a priori to be either convex or concave on each of a small number of regions of the function. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. For kernel estimates in Gaussian noise, the number of false change points is evaluated as a function of the smoothing parameter. To insure that the number of false convexity change points tends to zero, the smoothing level must be larger than is generically optimal for minimizing the mean integrated square error (MISE).
A two-stage estimator is proposed and shown to achieve the optimal rate of convergence of the MISE. In the first stage, the number and location of the change points is estimated using strong smoothing. In the second stage, a constrained smoothing spline fit is performed with the smoothing level chosen to minimize the MISE. The imposed constraint is that a single change point occurs in a region about each empirical change point from the first-stage estimate. This constraint is equivalent to the requirement that the third derivative of the second-stage estimate has a single sign in a small neighborhood about each first-stage change point. The change points from the second stage are first-stage change points, but need not be at the identical locations.