  Consider the problem of estimating a multivariate normal mean with a known
variance matrix, which is not necessarily proportional to the identity matrix.
The coordinates are shrunk directly in proportion to their variances in Efron
and Morris' (J. Amer. Statist. Assoc. 68 (1973) 117-130) empirical Bayes
approach, whereas inversely in proportion to their variances in Berger's (Ann.
Statist. 4 (1976) 223-226) minimax estimators. We propose a new minimax
estimator, by approximately minimizing the Bayes risk with a normal prior among
a class of minimax estimators where the shrinkage direction is open to
specification and the shrinkage magnitude is determined to achieve minimaxity.
The proposed estimator has an interesting simple form such that one group of
coordinates are shrunk in the direction of Berger's estimator and the remaining
coordinates are shrunk in the direction of the Bayes rule. Moreover, the
proposed estimator is scale adaptive: it can achieve close to the minimum Bayes
risk simultaneously over a scale class of normal priors (including the
specified prior) and achieve close to the minimax linear risk over a
corresponding scale class of hyper-rectangles. For various scenarios in our
numerical study, the proposed estimators with extreme priors yield more
substantial risk reduction than existing minimax estimators.
