Abstract: The value of a particular confidence set is traditionally measured by two quantities, its volume and probability of coverage. From a practical point of view, it is desirable to have a procedure that performs well against each measure. Such an approach, however, suffers in theory, for there is no one well-defined loss function which would place the problem in a true decision-theoretic setting. We derive necessary and sufficient conditions for minimax equivalence of the solution, using a linear combination loss function, to that of the component loss problem. Using this equivalence, it is possible to construct componentwise minimax estimators via the linear combination loss. We apply these results to estimation of a multivariate normal mean with unknown variance.
Key words and phrases: Confidence sets, multivariate normal density, minimax estimation.