Distributions of Risk Functions for the Pareto Model

Abstract

In statistical decision theory, the risk function quantifies the average performance of a decision over the sample space. The risk function, which depends on the parameter of the model, is often summarized by the Bayes risk, that is its expected value with respect to a design prior distribution assigned to the parameter. However, since expectation may not be an adequate synthesis of the random risk, we propose to examine the whole distribution of the risk function. Specifically, we consider point and interval estimation for the two parameters of the Pareto model. Using conjugate priors, we derive closed-form expressions for both the expected value and the density functions of the risk of each parameter under suitable losses. Finally, an application to wealth distribution is illustrated.