Revisiting the Lost Submarine Problem: A Decision Theoretic Approach
Anthony Almudevar
TL;DR
The paper reframes the classical lost submarine CI problem within a decision-theoretic framework, showing that the appropriate CI selection depends on the specified loss and operational goals. By deriving the distributional structure with $\hat{\theta}=M$ and an ancillary $V$, it analyzes four CI constructions (SD, NP, UMP, BC) and introduces an admissibility bound that constrains feasible intervals. It then demonstrates how a decision-theoretic formulation yields explicit optimal $b(v)$ functions under minimum-risk and minimum-search-effort criteria, including a threshold-based solution that concentrates probability mass on the shortest admissible intervals. The results highlight that no single CI is universally best; instead, the choice should align with the defined loss and practical constraints, clarifying when conventional intervals may be inadmissible or suboptimal. Overall, the framework provides a principled guide for CI selection in presence of ancillary information and explicit decision objectives, with $\hat{\theta}$ and $V$ playing central roles in shaping inference.
Abstract
This article includes a discussion of the ``lost submarine problem", following Morey \emph{et al} (2016). As the title of that paper suggests (\emph{The fallacy of placing confidence in confidence intervals}), the example is intended to illustrate the futility of relying on the confidence interval as a formal inference statement. In the view of this author, the misgivings expressed in Morey \emph{et al} (2016) can be resolved using a decision theoretic approach. While it is true that a variety of statistical methods lead to a variety of confidence intervals, once we precisely define their purpose, a single optimal choice emerges. Furthermore, distinct purposes lead to distinct optimal choices. Therefore, that a variety of procedures exist is an advantage rather than a liability.