Superefficient Estimation of Multivariate Trend

The question of recovering a multiband signal from noisy observations motivates a model in which the multivariate data points consist of an unknown deterministic trend Ξ observed with multivariate Gaussian errors. A cognate random trend model suggests affine shrinkage estimators Ξ̂A and Ξ̂B for Ξ, which are related to an extended Efron-Morris estimator. When represented canonically, Ξ̂A performs componentwise James-Stein shrinkage in a coordinate system that is determined by the data. Under the original deterministic trend model, Ξ̂A and its relatives are asymptotically minimax in Pinsker’s sense over certain classes of subsets of the parameter space. In such fashion, Ξ̂A and its cousins dominate the classically efficient least squares estimator. We illustrate their use to improve on the least squares fit of the multivariate linear model. AMS classification: 62H12, 62J05