On the optimality of coin-betting for mean estimation
Pith reviewed 2026-05-23 08:17 UTC · model grok-4.3
The pith
Coin-betting is optimal among e-variable frameworks for testing and estimating the mean of bounded random variables.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We establish the optimality of the coin-betting formulation among e-variable-based algorithmic frameworks for testing and estimating the (conditional) mean. As a consequence, we provide a direct and explicit characterisation of all valid e-variables and e-processes for this testing problem. In the language of classical statistical decision theory, we fully describe the set of all admissible e-variables and e-processes, and identify the corresponding minimal complete class.
What carries the argument
The notion of optimal classes for e-variables and e-processes, which is used both to prove optimality of coin-betting and to deliver the explicit characterization of all valid objects.
If this is right
- All valid e-variables for the mean-testing problem receive an explicit characterization.
- The full set of admissible e-variables and e-processes is described.
- The minimal complete class is identified in decision-theoretic terms.
- The optimality and characterization extend to estimation of conditional means.
Where Pith is reading between the lines
- The same optimality lens could be applied to derive characterizations for other bounded parameters.
- The explicit list may simplify construction of new sequential procedures that remain valid.
- Connections appear between the coin-betting objects and classical admissibility concepts in statistics.
Load-bearing premise
The introduced notion of optimal classes for e-variables and e-processes correctly captures the relevant optimality criteria for the mean-testing problem.
What would settle it
An explicit construction of a valid e-variable or e-process for mean testing that lies outside the coin-betting family yet satisfies the optimality criteria defined in the paper.
Figures
read the original abstract
We consider the problem of testing the mean of a bounded real random variable. We introduce a notion of optimal classes for e-variables and e-processes, and establish the optimality of the coin-betting formulation among e-variable-based algorithmic frameworks for testing and estimating the (conditional) mean. As a consequence, we provide a direct and explicit characterisation of all valid e-variables and e-processes for this testing problem. In the language of classical statistical decision theory, we fully describe the set of all admissible e-variables and e-processes, and identify the corresponding minimal complete class.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces a notion of optimal classes for e-variables and e-processes, then proves that the coin-betting construction is optimal among e-variable-based algorithmic frameworks for testing and estimating the (conditional) mean of bounded random variables. As a consequence it supplies an explicit characterization of all valid e-variables and e-processes for this problem, and in decision-theoretic language fully describes the admissible ones together with the corresponding minimal complete class.
Significance. If the central claims hold, the work supplies a complete, decision-theoretic characterization of admissible e-variables for bounded-mean testing. This is a substantive contribution to the theory of e-processes and sequential nonparametric inference, as it identifies a minimal complete class and thereby clarifies which procedures cannot be improved upon without additional assumptions.
minor comments (3)
- The abstract and introduction use the phrase 'optimal classes' without an immediate forward reference to its formal definition; adding a parenthetical pointer to the relevant section would improve readability.
- Notation for the betting fraction or wealth process is introduced in multiple places; a single consolidated table or displayed list of symbols would reduce the need to hunt for definitions.
- Several statements about 'all admissible e-variables' would benefit from an explicit remark on whether the characterization extends immediately to unbounded or vector-valued observations, even if only to state that it does not.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, for recognizing its contribution to the decision-theoretic characterization of admissible e-variables and e-processes, and for recommending minor revision. No specific major comments or points of criticism appear in the report.
Circularity Check
No significant circularity
full rationale
The paper introduces a novel decision-theoretic notion of optimal classes for e-variables and e-processes, then applies it to characterize admissible procedures and establish optimality of the coin-betting construction for bounded mean testing. No step reduces a claimed prediction or optimality result to a fitted quantity, self-referential definition, or load-bearing self-citation chain; the argument is framed as an application of classical admissibility and minimal complete class concepts to the newly defined framework, remaining self-contained without internal reduction to its own inputs.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The random variable is bounded (standard domain assumption for the coin-betting construction to be well-defined).
Reference graph
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discussion (0)
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