  Let y=A\beta+\epsilon, where y is an N\times1 vector of observations, \beta
is a p\times1 vector of unknown regression coefficients, A is an N\times p
design matrix and \epsilon is a spherically symmetric error term with unknown
scale parameter \sigma. We consider estimation of \beta under general quadratic
loss functions, and, in particular, extend the work of Strawderman [J. Amer.
Statist. Assoc. 73 (1978) 623-627] and Casella [Ann. Statist. 8 (1980)
1036-1056, J. Amer. Statist. Assoc. 80 (1985) 753-758] by finding adaptive
minimax estimators (which are, under the normality assumption, also generalized
Bayes) of \beta, which have greater numerical stability (i.e., smaller
condition number) than the usual least squares estimator. In particular, we
give a subclass of such estimators which, surprisingly, has a very simple form.
We also show that under certain conditions the generalized Bayes minimax
estimators in the normal case are also generalized Bayes and minimax in the
general case of spherically symmetric errors.
