The distribution of the risk function for interval estimation

In frequentist statistical decision theory, the average perfor- mance of an interval estimator is measured by its risk function, that is the expected value of the loss associated to the estimator. Risk functions depend on the unknown parameter of the model and might be summa- rized by the Bayes risk, their expected value with respect to a distribution assigned to the parameter. In this article we propose to study the entire distribution of the risk function induced by the prior and to complement the Bayes risk with additional summaries, such as the median and the mode, in order to define suitable criteria for sample size determination. We consider a standard loss, that is a monotone function of the interval length. We determine closed-form results for the Poisson-Gamma model and we illustrate an application to an oral health intervention study.