On the Order-Conditional Optimality of Gaffke's Bound

Let X = (X1, ldots, Xn) be a random vector from any Borel probability law on ℝ_+n. We revisit the problem of deriving a lower confidence bound (LCB) on a scalar parameter of that law. We recast classical work, beginning with Buehler, in purely probabilistic terms to form a more accessible and extensible framework. We then specialize the framework to the case where the components of X are independent. In this context, we prove that Gaffke's bound is Buehler optimal for the order that it induces with respect to the maximum marginal mean parameter: maxi ∈ [n] EQ[Xi], which reduces to the common mean when the Xi are independent and identically distributed. That is to say, no other valid LCB that orders samples in the same way as Gaffke's bound can improve on it with respect to this parameter.