A note on existence and construction of invariant loss functions

In a statistical decision problem, if the model is invariant under a transformation group, it is desirable or even compelling to apply equivariance for choosing a decision rule. However, formal equivariance also requires an invariant loss function. In this paper, we give a necessary and sufficient condition for the existence of invariant loss functions, and characterize all invariant loss functions, when the condition is satisfied. Analogous results for the more general case, where the quantity of inferential interest depends also on the observed data, are presented. We also discuss connections among our results and the equivariance literature and present some illustrative examples.