Sharp oracle inequalities for Least Squares estimators in shape restricted regression

Understand

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.