React Scatterplot Smoothers: Superefficiency through Basis Economy

Abstract

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. Applied to one- or higher-way layouts, the REACT method generates automatic scatterplot smoothers that compete well on standard datasets with the best fits 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 superefficiency relative to the least squares fit. This reduction in risk and the possible economy of the discrete cosine basis, of the orthogonal polynomial basis, or of a smooth basis that generalizes the discrete cosine basis are illustrated by fitting scatterplots drawn from the literature. Flexible monotone shrinkage of components rather than nested 1–0 shrinkage achieves a secondary decrease in risk that is visible in these examples. Pinsker bounds on asymptotic minimax risk for the estimation problem express the remarkable role of basis economy in reducing risk.

Key Words:

• Adaptation
• Asymptotic minimax
• Discrete cosine transform
• Linear model
• Minimum C_L
• Risk estimation
• Shrinkage
• Symmetric linear smoother.