A note on existence and construction of invariant loss functions

Haojin Zhou, Tapan K. Nayak*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)1335-1343
Number of pages9
Issue number6
Publication statusPublished - 25 Nov 2014
Externally publishedYes


  • equivariance
  • invariantly estimable
  • maximal invariant
  • target function
  • transformation group


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