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Optimal e-values for testing the mean of a bounded random variable against a composite alternative

T0 review · 0 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A single linear bet gives the optimal e-variable for bounded-mean tests.

desk verdict Solid, genuinely new result in e-value theory: the first explicit (RE)GROW characterizations for a non-AC problem, but the abstract's uniqueness claim is stronger than what is proved. read the letter →

arxiv 2601.11347 v2 pith:W6YF3ECG submitted 2026-01-16 math.ST stat.TH

Sebastian Arnold , Eugenio Clerico This is my paper · ORCID
classification math.STstat.TH MSC 62F0362G10
keywords e-valuesGROWREGROWboundedmeantestingcoin-bettinge-classcompositealternativeconvexenvelopenon-dominatedhypotheses
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives, for the first time outside the usual mutually-absolutely-continuous setting, the e-variables that are optimal for testing the mean of a bounded random variable under the GROW and REGROW criteria. Its central claim is that for a null hypothesis fixing the mean at µ0 and an alternative fixing it at µ1, the worst-case growth-optimal e-variable is the coin-betting function E(x)=1+α(x−µ0) with α=(µ1−µ0)/((µ0−a)(b−µ0)), and a unique REGROW e-variable of the same linear form exists, determined by an explicit equation. For the agnostic alternative that excludes only the null mean, the paper shows the GROW criterion degenerates to the constant e-variable 1, while REGROW still yields a non-trivial linear e-variable. These closed forms matter because they turn a previously abstract optimality criterion into concrete betting factors, and they show that REGROW can remain informative where the more pessimistic GROW criterion gives up.

What carries the argument

The load-bearing object is the coin-betting e-class, the one-parameter family E_α(x)=1+α(x−µ0), α∈[αmin,αmax], which are valid e-variables for the mean-µ0 null. A completeness theorem quoted from the authors' companion work says this family is minimal complete: every valid e-variable is pointwise dominated by some E_α. That reduction turns the infinite-dimensional search for a (RE)GROW e-variable into a one- or two-dimensional optimization over α and β. The second tool is the convex envelope f^c(x)=inf_{Q:E_Q[X]=x}E_Q[f], which by Lemma 1 is either f itself or the affine interpolation between endpoint values; this converts worst-case expectations over mean-constrained distributions into trac

What would settle it

Run a finite-dimensional optimization over all valid e-variables on a discrete grid of support points (e.g., X∈{−1,0,1} with null mean 0 and alternative mean 1/2), maximizing the worst-case expected log-growth, and compare the optimum to the paper's E_α* formula; any e-variable with a strictly larger worst-case value would falsify the completeness reduction on which the claim rests.

Watch

Extended reading notes

Core claim

The paper establishes explicit solutions to the (RE)GROW optimality problem for single-round tests of the mean of X∈[a,b]. For the point-null/point-alternative pair {E[X]=µ0} vs {E[X]=µ1}, the GROW e-variable exists and is E_{α*_GW}(x)=1+α*_GW(x−µ0), with α*_GW=(µ1−µ0)/((µ0−a)(b−µ0)). The REGROW e-variable also exists and is the same linear form with a unique parameter α*_RGW, determined for µ1>µ0 by solving F_{α*_RGW,αmax}(µ1)=G_{α*_RGW,α*_GW}(µ1); the case µ1<µ0 is symmetric. For one-sided hypotheses (mean ≤ µ0 vs mean > µ1), the GROW parameter has the same formula, while REGROW has the closed-form parameter given in the paper. For the agnostic alternative {E[X]≠µ0}, the paper shows that e

Load-bearing premise

The global optimality of every formula in the paper rests on the quoted completeness theorem (Proposition 1)—that every valid e-variable for the mean-µ0 null is pointwise dominated by some linear coin-betting variable 1+α(x−µ0)—which is stated as a corollary of the authors' companion work and not proved here.

Editorial extensions

If this is right

  • If correct, the paper gives explicit, closed-form e-variables for bounded-mean testing: a practitioner can set the betting parameter directly from µ0, µ1, and the support endpoints a,b.
  • For the agnostic alternative {mean ≠ µ0}, GROW-type worst-case optimality is vacuous, while REGROW still produces a non-trivial linear e-variable, sharpening the case for REGROW as the more useful criterion in this problem.
  • The optimal e-variables are always of the linear coin-betting form even though the null contains discrete, continuous, and singular distributions, providing a concrete minimal complete class for the stated testing problems.
  • The worst-case alternatives for the composite alternatives are all supported on at most two points—the boundary atoms δ_a and δ_b for the one-sided and agnostic problems—giving a simple picture of which alternatives are hardest to distinguish from the null.
  • For the Bernoulli special case (a=0, b=1), the paper's REGROW formulas reproduce the explicitly computed Bernoulli solutions, confirming consistency of the general theory with the simplest example.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the authors leave implicit is that the explicit linear factors multiply under independent observations, so they could be plugged into a product e-value to form an anytime-valid confidence sequence for the mean; the paper notes the product form but does not prove worst-case optimality for such a sequential strategy.
  • A testable extension is to run a finite-dimensional grid optimization over all valid e-variables on a discrete support and compare the worst-case expected log-growth with the paper's formula; this would directly check the completeness reduction in a concrete case.
  • If the quoted completeness theorem were extended to other linear constraints, the same convex-envelope proof scheme would yield explicit (RE)GROW e-variables for moment-constrained hypotheses; the paper does not pursue this, but the structure of the proof suggests it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper derives explicit GROW and REGROW e-variables for testing the mean of a bounded random variable against composite alternatives, in settings where the hypotheses are not mutually absolutely continuous. The null is the set of distributions with mean equal to (or bounded by) a fixed value, and the alternative is either a point alternative at a different mean, a one-sided interval of means, or the agnostic set of all other means. Using a reduction to the one-parameter coin-betting class (Proposition 1, borrowed from Clerico 2025a) and a c-envelope argument, the authors obtain: Theorem 1, the GROW parameter alpha*_GW = (mu1-mu0)/((mu0-a)(b-mu0)) and a REGROW parameter defined by the fixed-point equation (13); Theorem 2, the analogous one-sided result with closed form (18); and Theorem 3, the agnostic case where GROW is trivial and REGROW has parameter (19). The paper also identifies the worst-case alternative distributions and connects the results to the Bernoulli example.

Significance. The results are significant: they appear to provide the first explicit GROW/REGROW characterisations for non-dominated hypotheses in bounded-mean testing, an open direction noted in the e-value literature. The c-envelope lemma is proved self-containedly, and the derived formulas are simple and falsifiable; in the special case a=0, b=1 they reproduce the Bernoulli solutions of Example 1. The main external input, Proposition 1, is quoted from a paper co-authored by one of the present authors, but it is true and can be verified directly from the e-variable constraints, so I do not regard the reliance as circular. The principal imprecision is the abstract's claim of 'unique' e-values, which the manuscript's own statements do not justify in this non-AC setting.

minor comments (5)
  1. [Abstract; Theorems 1-3] The abstract says 'the unique e-values with optimal growth rate in the worst case'. In the present non-AC setting the maximisers are not unique: because the worst-case objective only depends on the e-variable through its values on the finite worst-case supports (e.g. {a,b} for GROW in (9), and also mu1 for REGROW), sufficiently small pointwise decreases of E_{alpha*} outside those supports preserve the e-variable property and the same worst-case growth. The theorems identify one optimal e-variable, not a unique one. Please replace 'unique' by 'admissible' or 'maximal' and avoid 'the GROW/REGROW e-variable' where uniqueness is not established.
  2. [Section 2, Proposition 1] The global optimality claims in Theorems 1-3 rest on Proposition 1, which is quoted from Clerico (2025a) and not proved here. Because the proposition is both load-bearing and self-cited, I recommend adding a short proof in the appendix or at least stating it as a lemma with proof. The argument is only a few lines: for a mixture p*delta_x+(1-p)*delta_y with mean mu0, the e-variable constraint gives R(x)+S(y) <= 0, yielding the required dominating E_alpha.
  3. [Appendix, proof of Theorem 1 around Eq. (15)] In the case beta in (0, alpha*_RGW), the assertion that beta -> G_{alpha*_RGW,beta}(mu1) has a unique minimizer at beta = alpha*_GW is stated without proof. This is a load-bearing step for (15). Please add the one-line justification: G_{alpha,beta}(mu1) equals L(log E_alpha)(mu1) - L(log E_beta)(mu1), and L(log E_beta)(mu1) is exactly the function maximized at alpha*_GW in the GROW computation.
  4. [Proof of Theorem 2] In the existence argument for tilde-alpha*_RGW, the expressions 'F_{alpha*_GW,alpha_max}(1)' and 'F_{alpha_max,alpha_max}(1)' should use the right endpoint b of the support, not 1; the support is [a,b]. This appears to be a leftover from the Bernoulli normalization a=0, b=1.
  5. [Proof of Theorem 1, Section 4.1] In the chain of inequalities for alpha > alpha*_RGW, the text says 'where we used the monotonicity of alpha -> G_{alpha,alpha_max}(mu1) in the second inequality'. The relevant function is alpha -> G_{alpha,alpha*_GW}(mu1), as in the existence argument. Please correct the typo.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the self-cited completeness theorem is independent of the (RE)GROW optimality claims, and the paper carries out the optimization explicitly.

full rationale

The derivation chain is not circular. The only external input imported by self-citation is Proposition 1 from Clerico (2025a), which says every e-variable for P={E_P[X]=μ0} is pointwise dominated by some coin-betting e-variable E_α(x)=1+α(x−μ0). Although this proposition is load-bearing and is authored by one of the present co-authors, it is not equivalent to the target result: it is a parameter-free completeness statement about the e-variable class, with assumptions that do not include GROW or REGROW optimality. It can also be verified directly from the e-variable constraint on two-point mixtures with mean μ0, so it constitutes independent support rather than a circular premise. Given Proposition 1 (and Corollary 1 for the one-sided null), the paper genuinely reduces the infinite-dimensional search to a one- or two-parameter optimization and solves it: for GROW it maximizes the concave function f^c_α(μ1), yielding the explicit formula (12); for REGROW it solves the balancing equations (13), (17), and (19). These are not fitted parameters renamed as predictions, nor are the optimal e-variables defined in terms of the quantities they are said to predict. The only noticeable imprecision is the abstract's phrase 'unique e-values': in this non-absolutely-continuous setting, pointwise modifications of E_α* outside the worst-case supports can preserve optimality, so the wording should be 'admissible' or 'maximal' rather than 'unique'. That is a correctness/overclaim issue, not a circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted constants and no new mathematical entities. Its load-bearing assumptions are external theorems from the e-value literature, one of which is by the present co-author Clerico; the rest of the analysis is a direct convex-optimization derivation.

assumptions (3)
  • domain assumption Proposition 1 (Clerico 2025a): every e-variable for P={mean=µ0} is pointwise dominated by a coin-betting e-variable E_α=1+α(x−µ0).
    Used in Theorems 1–3 to reduce the infinite-dimensional supremum over all e-variables to a one-parameter (GROW) or two-parameter (REGROW) optimization. This is a result from a prior paper by co-author Clerico and is not proved in the current manuscript.
  • domain assumption For a simple alternative {Q}, a GRO e-variable exists and is well defined under essentially no restriction on the null (Larsson et al. 2025b).
    Used in the definition of REGROW, which subtracts GRO(Q) for every Q in the composite alternative; without this existence result the REGROW objective is not defined.
  • domain assumption The set of e-variables for a union of null hypotheses is the intersection of the e-variable sets for the components (Larsson et al. 2025b).
    Used in the proof of Corollary 1 for the one-sided null {E_P[X]≤µ0}.

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Pith. "Pith review of Optimal e-values for testing the mean of a bounded random variable against a composite alternative." pith.science (2026). https://pith.science/paper/W6YF3ECG

@misc{pith2026260111347,
  author       = {Pith},
  title        = {Pith review of: Optimal e-values for testing the mean of a bounded random variable against a composite alternative},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W6YF3ECG}},
  note         = {Machine review of arXiv:2601.11347}
}
read the original abstract

We derive explicitly the e-values with optimal (relative) growth rate in the worst case for testing the mean of a bounded random variable, thereby providing the first application of the (RE)GROW quality criteria beyond the assumption of mutually absolutely continuous hypotheses for e-values originally proposed by Gr\"unwald et al. (2024). For both criteria, we explicitly characterise the alternatives that are most difficult to test against and show that they admit a meaningful interpretation. We give two important examples in which REGROW provides a powerful quality criterion to choose optimal e-variables whereas GROW leads to trivial solutions.

Figures

Figures reproduced from arXiv: 2601.11347 by the authors.

Figure 1
Figure 1. Visual intuition behind the Bernoulli example. The solid blue curve shows the optimal growth rate GRO(Qµ) = EQµ [log Zµ] = kl(µ, µ0) as a function of µ. For any fixed ε ∈ [0, 1], the e-power EQµ [log Zε] is affine in µ and coincides with the tangent to the blue curve at µ = ε. The green dotted line corresponds to the trivial e-variable Zµ0 = (1, 1)⊤, which has zero e-power for all µ. It is the only tangent that is e… view at source ↗
Figure 2
Figure 2. The parameter space I 2 µ0 with the regions where Fα,β is convex (red) or concave (yellow), for µ1 > µ0. By Lemma 1, F c α,β ≡ Fα,β in the red area and F c α,β ≡ Gα,β in the yellow area. For α > 0, the directional derivatives of the function β 7→ F c α,β(µ1) are represented as fine arrows. Note that β 7→ Fα⋆ RGW,β(µ1) has two local minima, achieved at the two black dots in the figure. Formally proving this fact is t… view at source ↗
Figure 3
Figure 3. The optimal α ⋆ GW from (12) and (16), and the optimal α ⋆ RGW and ˜α ⋆ RGW given from (13) and (17) as functions of µ1 (for the alternatives Q = {EQ[X] = µ1} and Q˜ = {EQ[X] > µ1}). Notably, α ⋆ GW ↓ 0 whereas α˜ ⋆ RGW > 0 as µ1 ↓ µ0, showing that GROW yields a trivial solution, while REGROW is meaningful for testing P˜ = {EQ[X] ≤ µ0} vs. Q˜ = {EQ[X] > µ0}. For µ1 < µ0 and the symmetric testing problem P˜′ = {EP [X… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Strong duality for the GROW criterion

    math.ST 2026-06 unverdicted novelty 7.0 of 10

    The GROW value for bounded e-variables equals the minimal relative entropy between weak-* closed convex hulls of arbitrary composite null and alternative sets.

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Works this paper leans on

2 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    On the optimality of coin-betting for mean estimation.Int

    Eugenio Clerico. On the optimality of coin-betting for mean estimation.Int. J. Approx. Reason., 187: 109550, 2025a. Eugenio Clerico. Optimal e-value testing for properly constrained hypotheses.Preprint arXiv:2412.21125, 2025b. Peter Gr¨ unwald, Rianne de Heide, and Wouter Koolen. Safe testing.J. R. Stat. Soc. Ser. B: Stat. Methodol., 86(5):1091–1128, 03

    arXiv

  2. [2024]

    The numeraire e-variable and reverse information projection.Ann

    Martin Larsson, Aaditya Ramdas, and Johannes Ruf. The numeraire e-variable and reverse information projection.Ann. Stat., 53(3):1015–1043, 2025a. Martin Larsson, Aaditya Ramdas, and Johannes Ruf. E-variables for hypotheses generated by constraints. Preprint arXiv:2504.02974, 2025b. Francesco Orabona and Kwang-Sung Jun. Tight concentrations and confidence ...

    arXiv

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