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Abstract

Suppose that the data is modeled as replicated realizations of a p-dimensional random vector whose mean µ is a trend of interest and whose covariance matrix sigma is unknown, positive defnite. REACT estimators for the trend involve transformation of the data to a new basis, estimating the risks of a class of candidate linear shrinkage estimators, and selecting the candidate estimator with smallest estimated risk. For Gaussian samples and quadratic loss, the maximum risks of REACT estimators proposed in this paper undercut that of the classically efficient sample mean vector. The superefficiency of the proposed estimators relative to the sample mean is most pronounced when the new basis provides an economical description of the vector sigma-1=2 µ, dimension p is not small, and sample size is much larger than p.A case study illustrates how vague prior knowledge may guide choice of a basis that reduces risk substantially. This research was supported at Universität Heidelberg by the Alexander von Humboldt Foundation and at Berkeley by National Science Foundation Grant DMS 99-70266. Dean Huber of the U.S. Forest Service in San Francisco provided the lumber-thickness data, both numbers and context.