In this paper, we consider the simultaneous estimation of Poisson means under the loss function L(m)(c)(theta, delta) = Sigma(i=1)(p) c(i) theta(-mi)(theta(i)-delta(i))(2), where m(1), ... , m(p) are known real numbers and c(1) , ... , c(p) are positive known constants. A necessary and sufficient condition for the loss functions to have estimators with bounded risk is given. In particular, we find that the naive estimator delta(0)(X) = 0 is the unique minimax admissible estimator under the loss L(m)(c) with ail m(i)'s equal to 2. The question of whether there exists any proper Bayes minimax estimator, under some loss functions, will be addressed in this paper. The estimators proposed by Ghosh et al. (Ann. Statist. 11 (1983), 351-376), etc., which dominate the usual estimator X, are shown to have unbounded risk functions, and truncated estimators having bounded risk are constructed. Further, the truncated estimators are shown to be admissible when it is known that the original estimators are admissible.
