Estimation for distributions with monotone likelihood ratio:case of vector valued parameter

An extension of a result about the estimation in Karlin and Rubin is given for the following case:The sample space, the parameter space and the decision space are subsets of a multi-dimensional Euclidean space, there is defined a suitable partial ordering in each of spaces, and a probability distribution has monotone likelihood ratio with respect to the partial orderings (see Ishii, 1976). In the special case when the loss function is quadratic a simple proof of a result in Karlin and Rubin is given. Stein's estimators are discussed as examples.