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Abstract

The question of recovering a multiband signal from noisy observationsmotivates a model in which the multivariate data points consist of anunknown deterministic trend Xi observed with multivariate Gaussian errors. A cognate random trend model suggests two affineshrinkage estimators for the deterministic trend, which arerelated to an extended Efron-Morris estimator. When represented canonically, the one affineshrinkage estimator performs componentwise James-Stein shrinkage in a coordinate system that is determined by the data. Under the originaldeterministic trend model, this affineshrinkage estimator and its relatives are asymptoticallyminimax in Pinsker's sense over certain classes of subsets of theparameter space. In such fashion, the affineshrinkage estimator and its cousins dominate theclassically efficient least squares estimator. We illustrate their use toimprove on the least squares fit of the multivariate linearmodel.