Median Loss Decision Theory

In this paper, we argue that replacing the expectation of the loss in statistical decision theory with the median of the loss leads to a viable and useful alternative to conventional risk minimization particularly because it can be used with heavy tailed distributions. We investigate three possible definitions for such medloss estimators and derive examples of them in several standard settings. We argue that the medloss definition based on the posterior is better than the other two definitions that do not permit optimization over large classes of estimators. We argue that median loss minimizing estimates often yield improved performance, have resistance to outliers as high as the usual robust estimates, and are resistant to the specific loss used to form them. In simulations with the posterior medloss formulation, we show how the estimates can be obtained numerically and that they can have better robustness properties than estimates derived from risk minimization.