Proper Bayes Minimax Multiple Shrinkage Estimation

Music City Center 

For the canonical problem of estimating a multivariate normal mean under squared error loss, minimax multiple shrinkage estimators adaptively shrink estimates towards multiple points and subspaces, thereby enhancing the scope of potential risk reduction while maintaining the safety guarantee of minimaxity. Motivated from a Bayesian point of view, the construction of such minimax estimators has relied, up to now, on using mixtures of improper priors yielding superharmonic marginals. Indeed, even just the existence of proper Bayes minimax multiple shrinkage estimators has been a challenging open problem, one that Bill and I worked on for the last 30 years. Happily, Bill ultimately came up with a novel unbiased estimate of risk argument to demonstrate, for the first time, the existence of such estimators, including the existence of proper Bayes minimax multiple shrinkage estimators based on mixtures of the Strawderman-type priors which he pioneered in 1971. Not only are such multiple shrinkage estimators automatically admissible, but they also allow for the interpretation of their adaptive mixture weights as valid posterior probabilities. (This work is joint with Pankaj Bhagwat and Bill Strawderman).