ABSTRACT
Invariance is a mathematical term for symmetry and in practice many statistical testing problems exhibit symmetries. The notion of invariance in statistical tests is of old origin. The unpublished work of Hunt and Stein (see Lehmann (1959)) toward the end of World War II has given this principle strong support as to its applicability and meaningfulness in the framework of the general class of all statistical tests. It is now established as a very powerful tool for proving the admissibility and minimax property of many statistical tests. It is a generally accepted principle that if a problem with a unique solution is invariant under a certain transformation, then the solution should be invariant under that transformation. The main reason for the strong intuitive appeal of an invariant decision procedure is the feeling that there should be or exists a unique best way of analyzing a collection of statistical information. Nevertheless in cases in which the use of an invariant procedure conflicts violently with the desire to make a correct decision with high probability or to have a small expected loss, the procedure must be abandoned.
