Constructing Bayes Minimax Estimators through Integral Transformations
Dominique Fourdrinier, William E. Strawderman, Martin T. Wells
TL;DR
This work develops a constructive framework for Bayes minimax estimation of the multivariate normal mean under quadratic loss by leveraging integral transforms with spherical priors. Central to the approach is translating minimaxity into a differential inequality for the $I$-transform of the prior-induced marginal, enabling explicit forms for the radial priors and their Bayes rules. The authors extend the method to variance mixtures of normals, providing a Laplace-transform-based condition that yields minimax estimators and presenting concrete examples (including Strawderman-type priors) with verifiable parameters. Overall, the paper offers a unified, transform-based pathway to robust Bayes minimax estimators, linking classical shrinkage priors with modern transform techniques and suggesting directions for extending these ideas to broader decision-theoretic problems.
Abstract
The problem of Bayes minimax estimation for the mean of a multivariate normal distribution under quadratic loss has attracted significant attention recently. These estimators have the advantageous property of being admissible, similar to Bayes procedures, while also providing the conservative risk guarantees typical of frequentist methods. This paper demonstrates that Bayes minimax estimators can be derived using integral transformation techniques, specifically through the \( I \)-transform and the Laplace transform, as long as appropriate spherical priors are selected. Several illustrative examples are included to highlight the effectiveness of the proposed approach.