Sharp oracle inequalities for Least Squares estimators in shape restricted regression

The performance of Least Squares (LS) estimators is studied in isotonic, unimodal and convex regression. In isotonic and unimodal regression, the LS estimator satisfies an adaptive risk bound of order $k/n$ up to logarithmic factors, where $k$ is the number of constant pieces of the true regression function. 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 in univariate convex regression holds for any design points. In isotonic and convex regression, our results have the form of sharp oracle inequalities that account for the model misspecification error.