REACT Fits to Linear Models and Scatterplots

REACT estimators for the mean of a linear model involve three steps: transforming the model to a canonical form that provides an economical representation of the unknown mean vector, estimating the risks of a class of candidate linear shrinkage estimators, and adaptively selecting the candidate estimator that minimizes estimated risk. When the mean vector is smooth, the desired canonical form of the linear model is achieved by constructing a smooth orthogonal basis for the regression space. Such a smooth basis for a complete, balanced one-way layout is asymptotically equivalent to the discrete cosine basis. Applied to one-or higher-way layouts, the REACT method generates automatic scatterplot smoothers that compete well on standard data sets with the best ts obtained by alternative techniques. Historical precursors to REACT include nested model selection, ridge regression, and nested principal component selection for the linear model. However, REACT's insistence on working with an economical basis greatly increases its supereeciency relative to the least squares t. A secondary improvement stems from REACT's use of exible monotone shrinkage rather than 0-1 shrinkage of components. Both improvements are demonstrated numerically on data sets and theoretically through Pinsker bounds for minimax risk in the estimation problem.