The Slope Heuristics in Heteroscedastic Regression

TL;DR

This paper investigates the optimal penalty in heteroscedastic nonparametric regression, establishing a minimal penalty, its relation to the optimal penalty, and proposing a hold-out method that achieves asymptotic optimality.

Contribution

It characterizes the minimal and optimal penalties for penalized least-squares model selection in heteroscedastic regression and introduces a hold-out procedure that is asymptotically optimal.

Findings

Existence of a minimal penalty where model selection fails

Optimal penalty is twice the minimal penalty

Hold-out penalization method is asymptotically optimal

Abstract

We consider the estimation of a regression function with random design and heteroscedastic noise in a nonparametric setting. More precisely, we address the problem of characterizing the optimal penalty when the regression function is estimated by using a penalized least-squares model selection method. In this context, we show the existence of a minimal penalty, defined to be the maximum level of penalization under which the model selection procedure totally misbehaves. The optimal penalty is shown to be twice the minimal one and to satisfy a non-asymptotic pathwise oracle inequality with leading constant almost one. Finally, the ideal penalty being unknown in general, we propose a hold-out penalization procedure and show that the latter is asymptotically optimal.