Model averaging: A shrinkage perspective

TL;DR

This paper investigates model averaging as a form of shrinkage estimation in linear regression, establishing theoretical connections, proposing a new Stein-type MA method, and demonstrating its asymptotic optimality with numerical validation.

Contribution

It introduces a novel Stein-type model averaging procedure based on blockwise Stein estimation, connecting MA with shrinkage estimators and proving its asymptotic optimality.

Findings

Optimal MA estimator is the best linear estimator with non-increasing weights.

Mallows MA can be viewed as a variation of positive-part Stein estimators.

Proposed Stein-type MA is asymptotically optimal when variance is known.

Abstract

Model averaging (MA), a technique for combining estimators from a set of candidate models, has attracted increasing attention in machine learning and statistics. In the existing literature, there is an implicit understanding that MA can be viewed as a form of shrinkage estimation that draws the response vector towards the subspaces spanned by the candidate models. This paper explores this perspective by establishing connections between MA and shrinkage in a linear regression setting with multiple nested models. We first demonstrate that the optimal MA estimator is the best linear estimator with monotonically non-increasing weights in a Gaussian sequence model. The Mallows MA (MMA), which estimates weights by minimizing the Mallows' $C_p$ over the unit simplex, can be viewed as a variation of the sum of a set of positive-part Stein estimators. Indeed, the latter estimator differs from the MMA only in that its optimization of Mallows' $C_p$ is within a suitably relaxed weight set. Motivated by these connections, we develop a novel MA procedure based on a blockwise Stein estimation. The resulting Stein-type MA estimator is asymptotically optimal across a broad parameter space when the variance is known. Numerical results support our theoretical findings. The connections established in this paper may open up new avenues for investigating MA from different perspectives. A discussion on some topics for future research concludes the paper.