  We find the local rate of convergence of the least squares estimator (LSE) of
a one dimensional convex regression function when (a) a certain number of
derivatives vanish at the point of interest, and (b) the true regression
function is locally affine. In each case we derive the limiting distribution of
the LSE and its derivative. The pointwise limiting distributions depend on the
second and third derivatives at 0 of the "invelope function" of the integral of
a two-sided Brownian motion with polynomial drifts. We also investigate the
inconsistency of the LSE and the unboundedness of its derivative at the
boundary of the domain of the covariate space. An estimator of the argmin of
the convex regression function is proposed and its asymptotic distribution is
derived. Further, we present some new results on the characterization of the
convex LSE that may be of independent interest.
