Statistical invariance principle ($\phi (gy)=\phi (y)$)

If sample transformations preserve the model and both hypotheses, the invariance principle restricts attention to tests whose decisions are unchanged by those transformations. Such tests are functions of a maximal invariant. In estimation, the analogous requirement is equivariance, accompanied by an invariant loss function so that risks transform compatibly. The principle respects symmetries; it does not by itself prove optimality among all unrestricted procedures.