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Smoothing in the Linear Model

Rudolf Beran, University of California, Berkeley Dept of Statistics

Geoffrey Watson's greatest influence has been through seminal work in three areas: linear models, nonparametric regression, and directional statistics. I would like to describe how the first two of these areas are coming together in recent work

REACT estimators for the mean of a Gaussian linear model use model-selection or shrinkage, ideas from signal-processing, and stable algorithms in statistical computing to exploit the superefficiency loophole in classical parametric information bounds. REACT estimators are adaptive symmetric linear smoothers that realize the benefits of Charles Stein's ideas on estimating high dimensional parameters. The acronym sketches the steps in the methodology: Risk Estimation and Adaptation after Coordinate Transformation.

If a linear combination of the first few vectors in the transformed regression basis closely approximates the unknown mean vector, then the asymptotic maximum risk of a monotone-shrinkage REACT estimator greatly undercuts that of the classically efficient least squares estimator. In experiments on scatterplots found in the smoothing literature, REACT fits draw remarkable benefit from the economy of some natural regression bases. These bases include orthogonal polynomials and the discrete cosine basis.

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