On the Distribution of the Random Risk Under Alternative Loss Functions

In standard statistical decision theory the average performance of a decision is quantified by the risk function, that depends on the parameter of the model. The risk function may be summarized by the Bayes risk, its average with respect to a prior distribution assigned to the parameter. However, the expected value might not be a good representative of risk’s randomness. We here propose to study the whole distribution of the risk function and to report additional summaries such as the median and the mode. For the specific set up of point estimation of a normal variance, we provide closed-form expressions for expected value, cumulative distribution and density functions of the risk under quadratic loss. A simulation-based approach is proposed when closed-form expressions of the distribution and its summaries are not available, for instance when alternative loss functions are considered. Finally, the problem of sample size determination is considered.