|
Abstract:
|
When decision theory was emerging as a promising new paradigm for the unification of Statistics, Charles Stein was to play a major role in shaping the landscape of relationships between admissibility, invariance and minimaxity. In some of his earliest work, the now legendary Hunt-Stein Theorem, he established algebraic invariance conditions under which the search for minimax procedures could be restricted to finding best invariant tests. With subtle counterexamples such as the non-minimaxity of the best invariant covariance estimator, he was then able to establish conclusively the algebraic inadequacy of the full linear group, that it was just too big for the theorem to apply. And for the landscape of admissibility, Stein early on established revealing necessary and sufficient conditions for admissibility. For the canonical problem of estimating a p-dimensional normal mean under squared error loss, he was determined to prove the admissibility of the best invariant estimator. Despite coming up with a beautiful new proof for p=2, he hit a brick wall for the case p = 3. Not to be defeated, he instead turned to prove its impossibility, coming up with a counterexample that ultimately paved the way for the now thriving enterprise of high dimensional shrinkage estimation. And on the seventh day, he rested!
|
