We investigate the problem of estimating the mean vector θ of a multivariate normal distribution with covariance matrix σ2Ip, when σ2 is unknown, and where the loss function is ‖δ−θ‖2σ2. We find a large class of (proper and generalized) Bayes minimax estimators of θ, and show that the result of Strawderman (1973) [8] is a special case of our result. Since a large subclass of the estimators found are proper Bayes, and therefore admissible, the class of admissible minimax estimators is substantially enlarged as well.

In Ghosh and Parsian (1981), the Lindley-Smith (1972) linear estimates of the multinonnal mean vector are shown to be generalized Bayes with respect to symmetric bowl shaped loss both when the common variance is known and unknown . The admissibility of such estimators is also proved in both these oases. The present paper gencralizes the findings of Ghosh and Parsian (1981) to a multiple regression model.

For the p-variate normal mean with known variances, the Lindley-Smith (1972) estimators are shown to be generalized Bayes under symmetric bowl shaped loss, and the admissibility of such estimators is proved. Also, when the variance-covariance matrix is unknown, generalized Bayes estimators are proposed under symmetric bowl shaped loss, and their admissibility is also proved.

For the p-variate Poisson mean, under the sum of weighted squared error losses, weights being reciprocals of variances, a class of proper Bayes minimax estimates dominating the usual estimate, namely the sample mean is produced. An example is given to illustrate this. The interrelation of our results with those of Clevenson and Zidek is pointed out.

Consider p independent distributions each belonging to the one parameter exponential family with distribution functions absolutely continuous with respect to Lebesgue measure. For estimating the natural parameter vector with p ≥ p0 (p0 is typically 2 or 3), a general class of estimators dominating the minimum variance unbiased estimator (MVUE) or an estimator which is a known constant multiple of the MVUE is produced under different weighted squared error losses. Included as special cases are some results of Hudson [13] and Berger [5]. Also, for a subfamily of the general exponential family, a class of estimators dominating the MVUE of the mean vector or an estimator which is a known constant multiple of the MVUE is produced. The major tool is to obtain a general solution to a basic differential inequality.