On the Order-Conditional Optimality of Gaffke's Bound

2026-07-24

AI Summary

This study addresses the construction of optimal lower confidence bounds for the maximum mean parameter of a vector of independent nonnegative random variables. By embedding Buehler’s classical approach within a purely probabilistic framework, the work establishes—for the first time—that the Gaffke bound possesses Buehler optimality under the sample ordering it induces in the case of independent components. This result confirms the inadmissibility of any improvement over the Gaffke bound within this setting, thereby extending classical confidence bound theory and providing a rigorous theoretical foundation for nonparametric lower-bound estimation.

Abstract

Let $X = (X_1, \ldots, X_n)$ be a random vector from any Borel probability law on $\mathbb{R}_+^n$. 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: $max_{i \in [n]} E_Q[X_i]$, which reduces to the common mean when the $X_i$ 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.

Problem

• lower confidence bound
• Buehler optimality
• Gaffke's bound
• maximum marginal mean
• order-conditional optimality

Innovation

• Buehler optimality
• Gaffke's bound
• lower confidence bound
• order-conditional optimality
• maximum marginal mean