The Admissibility of Certain Invariant Statistical Tests Involving a Translation Parameter
Introduction. The notion of invariance (or symmetry) has such strong intuitive appeal that many current statistical procedures have the invariance property and are in fact the best invariant procedures although they were proposed long before a general discussion of invariance was available. Hotelling [1], [2] and Pitman [3], [4] emphasized the invariant nature of certain tests and esti-mates. A general definition of the notion for the problem of testing hypotheses was given by Hunt and Stein who showed that in this case under severe restric-tions on the group of transformations an optimum invariant test is most stringent or more generally minimax with respect to an invariant loss function (see [5]). This result has been extended to more general decision problems and more general groups by Peisakoff [6]. However, these results do not prove admissibility of the procedures in question unless the group of transformations is compact.