  Given noisy data, function estimation is considered when the unknown function
is known a priori to consist of a small number of regions where the function is
either convex or concave. 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 ensure 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 occur 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 have a single sign in a small
neighborhood about each first-stage change point. The change points from the
second-stage are in a neighborhood of the first-stage change points, but need
not be at the identical locations.
