Analytic approximation of the interval of Bayes actions derived from a class of loss functions

Due to a lack of time or limited information, a precise determination of the loss function, and the computation of the associated Bayes action is not always possible. For these reasons, statisticians use mathematically tractable losses instead of subjective ones. We study the legitimacy of such losses l(0) with specific properties, by first constructing a class around l(0) which contains all plausible subjective losses, and then computing the set A(pi) of Bayes actions associated with this class. We derive an analytic approximation to A(pi). We apply these results when l(0) is the quadratic, LINEX and entropy loss. Finally, we investigate robustness characteristics using the analytic expression of A(pi).