Charles Stein and invariance: Beginning with the Hunt–Stein theorem

When statistical decision theory was emerging as a promising new paradigm, Charles Stein was to play a major role in the development of minimax theory for invariant statistical problems. In some of his earliest work with Gil Hunt, he set out to prove that, in problems where invariant procedures have constant risk, any best invariant test would be minimax among all tests. Although finding it not quite true in general, this led to the legendary Hunt–Stein theorem, which established the result under restrictive conditions on the underlying group of transformations. In decision problems invariant under such suitable groups, an overall minimax test was guaranteed to reside within the class of invariant procedures where it would typically be much easier to find. But when it did not seem possible to establish this result for invariance under the full linear group, he instead turned to prove its impossibility with counterexamples such as the nonminimaxity of the usual sample covariance estimator where the full linear group was just too big for the Hunt–Stein theorem to apply. Further explorations of invariance such as the sometimes problematic inference under a fiducial distribution, or the characterization of a best invariant procedure as a formal Bayes procedure under a right Haar prior, are further examples of the far reaching influence of Stein’s contributions to invariance theory.