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ABSTRACT 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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