Fitted value shrinkage

TL;DR

This paper introduces a penalized least-squares method for linear regression that maintains invariance under linear transformations of the design matrix, offering a computationally efficient alternative to ridge regression with comparable performance.

Contribution

The paper develops a new invariant penalized least-squares approach with methods for tuning parameter selection and analyzes its asymptotic properties, filling a gap in existing regression techniques.

Findings

Method performs similarly to ridge regression in experiments.

Proposed tuning parameter estimators have desirable asymptotic properties.

Invariance property is beneficial for categorical predictors and interactions.

Abstract

We propose a penalized least-squares method to fit the linear regression model with fitted values that are invariant to invertible linear transformations of the design matrix. This invariance is important, for example, when practitioners have categorical predictors and interactions. Our method has the same computational cost as ridge-penalized least squares, which lacks this invariance. We derive the expected squared distance between the vector of population fitted values and its shrinkage estimator as well as the tuning parameter value that minimizes this expectation. In addition to using cross validation, we construct two estimators of this optimal tuning parameter value and study their asymptotic properties. Our numerical experiments and data examples show that our method performs similarly to ridge-penalized least-squares.