Adaptive Procedures in Multiple Decision Problems and Hypothesis Testing

Abstract

Necessary and sufficient conditions for the existence of adaptive procedures for identification of one of several probability distributions or for testing a simple hypothesis against a simple alternative are obtained. By definition, adaptive procedures are required to exhibit the same asymptotic behavior for several parametric families as do the optimal (minimax) estimators for each of these families. The proofs are based on a multivariate version of Chernoff's theorem, providing asymptotic formulas for probabilities of large deviations for sums of i.i.d. random vectors. Some examples of adaptive procedures are considered, and the non-existence of such rules is established in certain situations.