A note on the distribution of risk functions for estimation in scale-exponential families

In parametric statistical decision theory, the performance of a decision function is evaluated in terms of its risk function, a quantity that depends on the parameter of the model. The risk function is typically summarized by the Bayes risk, its expected value with respect to a prior distribution assigned to the parameter. However, the study of the whole distribution of the risk function and the derivation of additional summaries (such as quantiles and the mode) may provide a better insight into the overall risk associated to the decision. In this article we consider models that are scale-exponential families. In the case of point or interval estimation we provide general closed-form expressions for cumulative distribution and density functions of the random risk along with some relevant summaries. We show how this unifying approach can be specialized to specific relevant models and also employed for the definition of sample size determination criteria.