Pierre C. Bellec

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. 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

Bounds on the Prediction Error of Penalized Least Squares Estimators with Convex Penalty

n^{-2/3}

A sharp oracle inequality for Graph-Slope

as well as a parametric rate of order

Slope meets Lasso: Improved oracle bounds and optimality

k/n

Sharp oracle bounds for monotone and convex regression through aggregation

up to logarithmic factors, where

Optimal bounds for aggregation of affine estimators

k

Towards the study of least squares estimators with convex penalty

is the number of constant pieces of the true parameter. In univariate convex regression, the LS estimator satisfies an adaptive risk bound of order

Adaptive confidence sets in shape restricted regression

q/n

Optimistic lower bounds for convex regularized least-squares

up to logarithmic factors, where

Aggregation of supports along the Lasso path

q