Proper Bayes Minimax Multiple Shrinkage Estimation
ED GEORGE – UNIVERSITY OF PENNSYLVANIA
ABSTRACT
For the canonical problem of estimating a multivariate normal mean under squared error loss, minimax multiple shrinkage estimators adaptively shrink estimates towards the most promising of multiple prespecified targets, dramatically enhancing the scope of potential risk reduction while maintaining the global protection of minimaxity. Motivated from a Bayesian point of view, the construction of such estimators has relied on using mixtures of improper superharmonic priors to guarantee minimaxity. Indeed, until now, even just the mere existence of proper Bayes minimax multiple shrinkage estimators had remained a challenging open problem, one that Bill Strawderman and I struggled with together for over 30 years. Happily, Bill ultimately came up with a novel unbiased-estimate-of-risk decomposition which paved the way for the construction of such estimators, including the construction of proper Bayes minimax multiple shrinkage estimators driven by mixtures of the Strawderman-type priors which he pioneered in 1971. Not only are such proper Bayes minimax multiple shrinkage estimators automatically admissible, but they are probabilistically coherent, allowing for the interpretation of their adaptive mixture weights as valid posterior probabilities in contrast to their improper mixture prior counterparts.
This work is joint with Pankaj Bhagwat and Bill Strawderman.