Manifold-valued Tuning Parameters in Regularized Estimation of Multivariate Means
Rudolf Beran (May 24, 2012)
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A multivariate k-way layout consists of observations with error on an array of vector-valued means, each of which is an unknown function of k real-valued covariates. Any decomposition of these vector means into a sum of orthogonal projections induces least squares submodel fits that serve as candidate estimators of the mean vectors. MANOVA submodel fits, nested polynomial regression fits, or mixed combinations of both strategies illustrate classically. This talk describes penalized least squares estimators of the multivariate means in which the penalty terms are weighted through manifold-valued tuning parameters. Data-based selection of the tuning parameters yields estimators that dominate asymptotically those that arise from submodel fitting. In the special case of a complete balanced multivariate k-way layout, the proposed regularized estimators are linked to multiple Efron-Morris affine shrinkage. In unbalanced designs, the regularized estimators define a powerful generalization of affine shrinkage.