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