  The performance of Least Squares (LS) estimators is studied in isotonic,
unimodal and convex regression. Our results have the form of sharp oracle
inequalities that account for the model misspecification error. In isotonic and
unimodal regression, the LS estimator achieves the nonparametric rate
$n^{-2/3}$ as well as a parametric rate of order $k/n$ up to logarithmic
factors, where $k$ is the number of constant pieces of the true parameter.
  In univariate convex regression, the LS estimator satisfies an adaptive risk
bound of order $q/n$ up to logarithmic factors, where $q$ is the number of
affine pieces of the true regression function. This adaptive risk bound holds
for any design points. While Guntuboyina and Sen (2013) established that the
nonparametric rate of convex regression is of order $n^{-4/5}$ for equispaced
design points, we show that the nonparametric rate of convex regression can be
as slow as $n^{-2/3}$ for some worst-case design points. This phenomenon can be
explained as follows: Although convexity brings more structure than
unimodality, for some worst-case design points this extra structure is
uninformative and the nonparametric rates of unimodal regression and convex
regression are both $n^{-2/3}$.
