Optimal actions in problems with convex loss functions

Researches in Bayesian sensitivity analysis and robustness have mainly dealt with the computation of the range of some quantities of interest when the prior distribution varies in some class. Recently, researchers' attention turned to the loss function, mostly to the changes in posterior expected loss and optimal actions. In particular, the search for optimal actions under classes of priors and/or loss functions has lead, as a first approximation, to consider the set of nondominated actions. However, this set is often too big to take it as the solution of the decision problem and some criteria are needed to choose an optimal alternative within the nondominated set. Some authors recommended to choose the conditional ?-minimax or the posterior regret ?-minimax alternative within the set of all possible alternatives. These criteria are quite controversial since they could lead to actions with huge relative increase in posterior expected loss with respect to Bayes actions. To overcome such drawback, we propose a new method, based on the smallest relative error, to choose the least sensitive action and to discriminate alternatives within the nondominated set when the decision maker is interested in diminishing the relative error. We study how to compute the least sensitive action when we consider classes of convex loss functions. Furthermore, we obtain its relation with other proposed solutions: nondominated, minimax and posterior regret minimax actions. We conclude the paper with an example on the estimation of the mean of a Poisson distribution. © 2008 Elsevier Inc. All rights reserved.