Degrees of Freedom for Piecewise Lipschitz Estimators

TL;DR

This paper derives a novel representation of degrees of freedom for estimators with discontinuities, such as those in variable selection methods like lasso-OLS and best subset selection, enabling better risk estimation and model selection.

Contribution

It introduces a new degrees of freedom formula applicable to discontinuous estimators, extending Stein's lemma, and demonstrates its use in risk estimation for lasso-OLS and subset selection.

Findings

New degrees of freedom representation for discontinuous estimators

Effective risk estimation for lasso-OLS using the solution path

Improved variable selection via risk estimates in simulations

Abstract

A representation of the degrees of freedom akin to Stein's lemma is given for a class of estimators of a mean value parameter in $\mathbb{R}^n$. Contrary to previous results our representation holds for a range of discontinues estimators. It shows that even though the discontinuities form a Lebesgue null set, they cannot be ignored when computing degrees of freedom. Estimators with discontinuities arise naturally in regression if data driven variable selection is used. Two such examples, namely best subset selection and lasso-OLS, are considered in detail in this paper. For lasso-OLS the general representation leads to an estimate of the degrees of freedom based on the lasso solution path, which in turn can be used for estimating the risk of lasso-OLS. A similar estimate is proposed for best subset selection. The usefulness of the risk estimates for selecting the number of variables is demonstrated via simulations with a particular focus on lasso-OLS.