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’ 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’ 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.
Acknowledgments
The first version of this paper was completed during the author’s visit to the University of Minnesota in 2022. The author would like to thank Professor Yang Li for his unreserved support during the author’s visit to the University of Minnesota. The author also thanks Professor Yuhong Yang for helpful comments on an earlier version of this manuscript. Insightful and constructive comments from two anonymous referees are greatly appreciated.
Citation
Jingfu Peng. "Model averaging: A shrinkage perspective." Electron. J. Statist. 18 (2) 3535 - 3572, 2024. https://doi.org/10.1214/24-EJS2282
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