🤖 AI Summary
This work addresses Bayesian shrinkage estimation for the mean of a multivariate normal distribution under multiple prior target information. It proposes a general construction that, for the first time, explicitly yields a proper Bayes minimax multiple-shrinkage estimator. The method leverages a mixture structure based on square-root-harmonic marginal densities to adaptively shrink toward the more favorable among several prespecified targets, thereby overcoming limitations of existing approaches that rely on either a single target or improper priors. The resulting estimator possesses minimaxity, uniformly dominating the maximum likelihood estimator in terms of risk and achieving substantial risk reduction over a broad region of the parameter space, thus enhancing both robustness and efficiency.
📝 Abstract
For the canonical problem of estimating a multivariate normal mean under squared error loss, we demonstrate, for the first time, the existence of proper Bayes minimax multiple shrinkage estimators by introducing a general approach for their explicit construction. As opposed to minimax shrinkage estimators that shrink towards a single prespecified target, minimax multiple shrinkage estimators adaptively shrink towards the more promising of a set of prespecified targets, substantially increasing the region of potential risk reduction while maintaining the protection of always being at least as good as the maximum likelihood estimator. These estimators are particularly useful in practice as they address the challenge of selecting a minimax shrinkage estimator when prior information suggests more than one viable shrinkage target to choose from. In contrast to previous formal Bayes minimax multiple shrinkage estimators, which were built on mixtures of superharmonic marginals, these proper Bayes minimax multiple shrinkage estimators are obtained via mixtures of square-root superharmonic marginals. Examples of such proper Bayes minimax multiple shrinkage estimators include an adaptive convex combination of the classical Strawderman shrinkage estimators.