Recent theory for shrinkage estimators, techniques from signal-processing, and effective algorithms for computing orthonormal bases now make it possible to exploit the superefficiency loophole in classical information bounds for estimation. In linear regression, if the first few vectors in the regression basis closely approximate the unknown mean vector, then the risk of an estimator that shrinks to zero those regression coefficients associated with the unimportant basis vectors can be much smaller than the risk of the least squares estimator. Such shrinkage estimators, which are particular symmetric linear smoothers, realize the benefits of C. Stein's and M. S. Pinsker's pioneering ideas on estimation of high- or infinite-dimensional parameters. Specific goals of the research are: (a) to construct and interpret confidence sets centered at a superefficient fit; (b) to handle, through multiple shrinkage, cases where the chosen basis is sparse but not well-ordered; (c) to develop within- and between-observation shrinkage techniques to handle the multivariate linear model; (d) to draw on relations with signal-processing techniques that use the discrete cosine basis or wavelet bases.
Regression models fitted by the method of least squares are widely used in scientific research and other disciplines to establish quantitative relationships within sets of data. Studies related to the program on Environment and Global Change and to the program on Manufacturing are examples. The broad goal of the proposed research is to improve the reliability of these fitted relationships by replacing least squares with better adaptive linear smoothers. Recent statistical theory supports the general feasibility of this project. How to realize what is possible in theory is the essence of the work. The author's REACT method, described with references in the proposal, is a practical technique for regression with one response variable that demonstrates real-world improvements over least squares fits. REACT competes well with current nonparametric regression methods while offering certain advantages, such as built-in diagnostics that indicate the quality of the fit. The proposed research will extend REACT methods to finding relationships among sets of variables and will develop practical methods for assessing the uncertainty of the estimated relationships. Least squares regression, a standard function in modern statistical packages and spreadsheets, is widely used in data-analysis. This circumstance provides strong motivation for improving on least squares.
