**Statistical Computing '99 - Schloß Reisensburg**

## Least Squares, Statistical Computing, and REACT Fits

### Rudolf Beran, Berkeley

REACT estimators for the mean
of a Gaussian linear model use
model-selection, shrinkage, ideas
from signal-processing, and stable
algorithms in statistical computing
to exploit the superefficiency
loophole in classical parametric
information bounds. REACT methods may be
also be viewed as smoothing
techniques. 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 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.

Which features of a REACT
estimator are not necessarily present in
the true mean vector? Which
features of the true mean vector have been
smoothed out by the REACT fit?
Constructing extremal members of a
confidence set centered at the REACT
estimator begins to address these
questions.

31. Statistical Computing '99