Constrained forms of statistical minimax :

A fundamental quantity in statistical decision theory is the notion of the minimax risk as5 sociated with an estimation problem. It is based on a saddlepoint problem, in which nature plays the 6 role of adversary in choosing the underlying problem instance, and the statistician seeks an estimator 7 with good properties uniformly over a class of problem instances. We argue that in many modern 8 estimation problems arising in the mathematical sciences, the classical notion of minimax risk suf9 fers from a significant deficiency: to wit, it allows for all possible estimators, including those with 10 prohibitive computational costs, unmanageable storage requirements, or other undesirable properties. 11 Accordingly, we introduce some refinements of minimax risk based on imposing additional constraints 12 on the sets of possible estimators. We illustrate this notion of constrained statistical minimax via three 13 vignettes, based on restrictions involving computation, communication, and privacy, respectively. 14 Mathematics Subject Classification (2010). Primary 62Cxx; Secondary 68W40. 15