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arxiv: 2606.06332 · v1 · pith:ST3FMDDLnew · submitted 2026-06-04 · 🧮 math.ST · stat.ME· stat.ML· stat.TH

Bentkus-type asymptotic e-values

Diego Martinez-Taboada , Ben Chugg , Aaditya Ramdas This is my paper

Pith reviewed 2026-06-27 23:21 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.MLstat.TH
keywords asymptotic e-valuesBentkus inequalitiesmissing factorpost-hoc inferencemultiple testingconcentration inequalitiese-values
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The pith

Bentkus-type asymptotic e-values remove the missing factor that made earlier versions overly conservative.

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

The paper introduces Bentkus-type asymptotic e-values by adapting near-optimal concentration inequalities from Bentkus. These e-values eliminate the missing factor, a scaling inefficiency that made existing asymptotic e-values overly conservative. This change produces sharper inference overall. Readers would care because the improvement yields tighter post-hoc confidence intervals and higher rejection rates in multiple testing while preserving asymptotic validity.

Core claim

Drawing on Bentkus's framework of near-optimal concentration inequalities, the authors construct Bentkus-type asymptotic e-values and prove that they eliminate the missing factor without introducing new asymptotic inefficiencies or validity issues. They further show both theoretically and empirically that these e-values deliver sharper inference than existing alternatives.

What carries the argument

Bentkus-type asymptotic e-values, which adapt Bentkus's near-optimal concentration inequalities to the construction of e-values in order to remove the missing factor.

If this is right

  • Tighter post-hoc confidence intervals result from the improved scaling.
  • Higher rejection rates occur in multiple testing procedures.
  • Sharper inference holds compared to existing asymptotic e-values.
  • Validity is maintained while asymptotic efficiency improves.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same adaptation technique could be tested on other asymptotic statistics that currently suffer from similar scaling losses.
  • Finite-sample behavior might be checked by comparing interval widths across a range of sample sizes not covered in the paper.
  • The method might connect to other uses of concentration inequalities for constructing sequential tests.

Load-bearing premise

Bentkus's near-optimal concentration inequalities can be directly adapted to asymptotic e-values in a way that removes the missing factor without creating new validity or efficiency problems.

What would settle it

A concrete numerical example or simulation in which Bentkus-type e-values retain the same scaling inefficiency as prior asymptotic e-values would show that the missing factor has not been eliminated.

Figures

Figures reproduced from arXiv: 2606.06332 by Aaditya Ramdas, Ben Chugg, Diego Martinez-Taboada.

Figure 1
Figure 1. Figure 1: Evaluation of Uδ,∞(λ) and Uδ,α(λ) for α ∈ {0, 1, 2, 3, 4, 5} and δ ∈ {0.1, 0.01, 0.001}. This should be interpreted as follows: Exponential e-values (α = ∞) and large-α Bentkus-type e-values are more robust but also more conservative. The width of the interval is smaller for values λ that are far from the minimizers, but larger otherwise. Consequently, the best choice of asymptotic e-value depends on the t… view at source ↗
Figure 2
Figure 2. Figure 2: Threshold functions Uδ,α(λ) for α ∈ {1, 2, ∞} and δ ∈ {0.1, 0.05, 0.01}. The minima for α = 1 and α = 2 lie below the minimum for α = ∞ at every displayed level, reflecting the near-optimality established in Theorem 4.3. incurs a hedging penalty against an optimal oracle, the threshold regret R(δ) is strictly bounded by a term proportional to 1/G−1 (δ). In particular, the theoretical cost of remaining agno… view at source ↗
read the original abstract

Asymptotic e-values are emerging as a powerful alternative to asymptotic p-values, particularly in post-hoc inference and multiple testing, where significance levels may be data-dependent. Existing asymptotic e-values, however, suffer from the ``missing factor,'' a scaling inefficiency resulting in overly conservative inference. Drawing on the framework of near-optimal concentration inequalities developed by Bentkus in the 2000s, we introduce Bentkus-type asymptotic e-values and prove that they successfully eliminate the missing factor. We also demonstrate both theoretically and empirically that Bentkus-type e-values consistently deliver sharper inference than existing alternatives, leading to tighter post-hoc confidence intervals and higher rejection rates in multiple testing procedures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces Bentkus-type asymptotic e-values constructed by adapting Bentkus' near-optimal concentration inequalities. It claims to prove that these e-values eliminate the 'missing factor' inefficiency of prior asymptotic e-values (ensuring E[e_n] ≤ 1 + o(1) under the null) while delivering sharper inference, with theoretical arguments and empirical results showing tighter post-hoc confidence intervals and higher power in multiple testing.

Significance. If the central claims hold, the work provides a concrete improvement to asymptotic e-value constructions by leveraging established, near-optimal concentration results from Bentkus. This addresses a documented scaling inefficiency without introducing new asymptotic costs, which would strengthen applications in data-dependent inference and multiple testing. The grounding in external inequalities and the combination of proofs with empirical validation are strengths.

minor comments (3)
  1. The abstract and introduction should include a precise definition or equation for the 'missing factor' (e.g., the explicit scaling term in existing e-values) to make the improvement claim immediately verifiable.
  2. Empirical sections would benefit from explicit statements of data exclusion rules, simulation parameters, and verification that the reported rejection rates and interval widths are computed under the same null distributions as the theoretical comparisons.
  3. Notation for the Bentkus-type e-value construction should be introduced with a numbered display equation early in the methods section to facilitate cross-references in the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on Bentkus-type asymptotic e-values, including the recognition that they eliminate the missing factor and deliver sharper inference in post-hoc and multiple testing settings. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper adapts external Bentkus concentration inequalities (from the 2000s, independent of the authors) to construct asymptotic e-values that eliminate the missing factor. The central claims rest on proofs of validity (E[e_n] ≤ 1 + o(1)) and sharpness comparisons that draw on established e-value theory and these external inequalities rather than self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. No derivation step reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Limited information from abstract only; main reliance is on applicability of prior concentration inequalities.

axioms (1)
  • domain assumption Bentkus near-optimal concentration inequalities apply in the asymptotic e-value construction setting
    Paper draws directly on this 2000s framework to eliminate the missing factor.

pith-pipeline@v0.9.1-grok · 5638 in / 1201 out tokens · 50791 ms · 2026-06-27T23:21:50.886336+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

52 extracted references · 12 canonical work pages

  1. [1]

    Lithuanian Mathematical Journal , volume=

    On domination of tail probabilities of (super) martingales: explicit bounds , author=. Lithuanian Mathematical Journal , volume=. 2006 , publisher=

    2006

  2. [2]

    International Conference on Machine Learning , pages=

    Near-Optimal Confidence Sequences for Bounded Random Variables , author=. International Conference on Machine Learning , pages=. 2021 , organization=

    2021

  3. [3]

    A remark on the inequalities of

    Bentkus, V , journal=. A remark on the inequalities of

  4. [4]

    arXiv preprint arXiv:2412.20542 , year=

    On the Missing Factor in Some Concentration Inequalities for Martingales , author=. arXiv preprint arXiv:2412.20542 , year=

    work page arXiv

  5. [5]

    arXiv preprint arXiv:2603.08002 , year=

    Post-Hoc Large-Sample Statistical Inference , author=. arXiv preprint arXiv:2603.08002 , year=

    work page arXiv

  6. [6]

    Bentkus, Vidmantas , journal=. On. 2004 , publisher=

    2004

  7. [7]

    Lecture Notes-Monograph Series , pages=

    Binomial upper bounds on generalized moments and tail probabilities of (super) martingales with differences bounded from above , author=. Lecture Notes-Monograph Series , pages=. 2006 , publisher=

    2006

  8. [8]

    Asymptotic and compound e-values: multiple testing and empirical

    Ignatiadis, Nikolaos and Wang, Ruodu and Ramdas, Aaditya , journal=. Asymptotic and compound e-values: multiple testing and empirical

  9. [9]

    Foundations and Trends

    Hypothesis testing with e-values , author=. Foundations and Trends. 2025 , publisher=

    2025

  10. [10]

    arXiv preprint arXiv:2205.06812 , year=

    Principal-agent hypothesis testing , author=. arXiv preprint arXiv:2205.06812 , year=

    work page arXiv

  11. [11]

    arXiv preprint arXiv:2409.11134 , year=

    E-Values for exponential families: the general case , author=. arXiv preprint arXiv:2409.11134 , year=

    work page arXiv

  12. [12]

    arXiv preprint arXiv:2404.19465 , year=

    Optimal e-values for exponential families: The simple case , author=. arXiv preprint arXiv:2404.19465 , year=

    work page arXiv

  13. [13]

    Proceedings of the National Academy of Sciences , volume=

    Universal inference , author=. Proceedings of the National Academy of Sciences , volume=. 2020 , publisher=

    2020

  14. [14]

    arXiv preprint arXiv:2407.15733 , year=

    Admissible online closed testing must employ e-values , author=. arXiv preprint arXiv:2407.15733 , year=

    work page arXiv

  15. [15]

    Nonasymptotic Heavy-Tailed Mean Estimation in Smooth

    Whitehouse, Justin and Chugg, Ben and Martinez-Taboada, Diego and Ramdas, Aaditya , journal=. Nonasymptotic Heavy-Tailed Mean Estimation in Smooth. 2026 , publisher=

    2026

  16. [16]

    arXiv preprint arXiv:2509.02517 , year=

    Bringing closure to false discovery rate control: A general principle for multiple testing , author=. arXiv preprint arXiv:2509.02517 , year=

    work page arXiv

  17. [17]

    The missing factor in

    Talagrand, Michel , booktitle=. The missing factor in

  18. [18]

    Gr. Beyond. Proceedings of the National Academy of Sciences , volume=. 2024 , publisher=

    2024

  19. [19]

    International Journal of Approximate Reasoning , pages=

    On admissibility in post-hoc hypothesis testing , author=. International Journal of Approximate Reasoning , pages=. 2026 , publisher=

    2026

  20. [20]

    arXiv preprint arXiv:2312.08040 , year=

    Post-hoc Hypothesis Testing and the Post-hoc p -value , author=. arXiv preprint arXiv:2312.08040 , year=

    work page arXiv

  21. [21]

    Electronic Journal of Statistics , volume=

    Post-selection inference for e-value based confidence intervals , author=. Electronic Journal of Statistics , volume=. 2024 , publisher=

    2024

  22. [22]

    The missing factor in

    Fan, Xiequan and Grama, Ion and Liu, Quansheng , journal=. The missing factor in

  23. [23]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    False discovery rate control with e-values , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2022 , publisher=

    2022

  24. [24]

    arXiv preprint arXiv:2603.16213 , year=

    Equivalence testing with data-dependent and post-hoc equivalence margins , author=. arXiv preprint arXiv:2603.16213 , year=

    work page arXiv

  25. [25]

    Journal of the Royal Statistical Society: Series B (Methodological) , volume=

    Controlling the false discovery rate: a practical and powerful approach to multiple testing , author=. Journal of the Royal Statistical Society: Series B (Methodological) , volume=. 1995 , publisher=

    1995

  26. [26]

    Annals of statistics , pages=

    The control of the false discovery rate in multiple testing under dependency , author=. Annals of statistics , pages=. 2001 , publisher=

    2001

  27. [27]

    Empirical

    Martinez-Taboada, Diego and Ramdas, Aaditya , journal=. Empirical

  28. [28]

    , author=

    On normal domination of (super) martingales. , author=. Electronic Journal of Probability [electronic only] , volume=. 2006 , publisher=

    2006

  29. [29]

    The Annals of Probability , volume=

    When is the Student t -statistic asymptotically standard normal? , author=. The Annals of Probability , volume=. 1997 , publisher=

    1997

  30. [30]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    The moment generating function of the truncated multi-normal distribution , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 1961 , publisher=

    1961

  31. [31]

    1994 , publisher=

    Continuous univariate distributions, volume 1 , author=. 1994 , publisher=

    1994

  32. [32]

    2006 , publisher=

    Handbook of mathematical functions: with formulas, graphs, and mathematical tables , author=. 2006 , publisher=

    2006

  33. [33]

    NeuroImage , volume=

    Thresholding of statistical maps in functional neuroimaging using the false discovery rate , author=. NeuroImage , volume=. 2002 , publisher=

    2002

  34. [34]

    Proceedings of the National Academy of Sciences , volume=

    Statistical significance for genomewide studies , author=. Proceedings of the National Academy of Sciences , volume=. 2003 , publisher=

    2003

  35. [35]

    Genome Biology , volume=

    A practical guide to methods controlling false discoveries in computational biology , author=. Genome Biology , volume=. 2019 , publisher=

    2019

  36. [36]

    False discovery rate revisited:

    Chumbley, Justin R and Friston, Karl J , journal=. False discovery rate revisited:. 2009 , publisher=

    2009

  37. [37]

    Australian Journal of Statistics , volume=

    A theorem on products of random variables, with application to regression , author=. Australian Journal of Statistics , volume=. 1981 , publisher=

    1981

  38. [38]

    Risks , volume=

    An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality , author=. Risks , volume=. 2014 , publisher=

    2014

  39. [39]

    arXiv preprint arXiv:2602.13465 , year=

    Intrinsic dimension concentration inequalities for self-adjoint operators , author=. arXiv preprint arXiv:2602.13465 , year=

    work page arXiv

  40. [40]

    Sharp empirical

    Martinez-Taboada, Diego and Ramdas, Aaditya , journal=. Sharp empirical

  41. [41]

    Vector-valued self-normalized concentration inequalities beyond sub-

    Martinez-Taboada, Diego and Gonzalez, Tomas and Ramdas, Aaditya , booktitle=. Vector-valued self-normalized concentration inequalities beyond sub-. 2026 , organization=

    2026

  42. [42]

    arXiv preprint arXiv:2512.16759 , year=

    de Roos, Dante and Chugg, Ben and Gr. arXiv preprint arXiv:2512.16759 , year=

    work page arXiv

  43. [43]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Safe testing , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2024 , publisher=

    2024

  44. [44]

    arXiv preprint arXiv:2503.13050 , year=

    E-values expand the scope of conformal prediction , author=. arXiv preprint arXiv:2503.13050 , year=

    work page arXiv

  45. [45]

    Annales de l'IHP Probabilit

    Challenging the empirical mean and empirical variance: a deviation study , author=. Annales de l'IHP Probabilit

  46. [46]

    The Annals of Statistics , volume=

    E-values: Calibration, combination and applications , author=. The Annals of Statistics , volume=. 2021 , publisher=

    2021

  47. [47]

    The Annals of Statistics , volume=

    Time-uniform, nonparametric, nonasymptotic confidence sequences , author=. The Annals of Statistics , volume=. 2021 , publisher=

    2021

  48. [48]

    Time-uniform

    Howard, Steven R and Ramdas, Aaditya and McAuliffe, Jon and Sekhon, Jasjeet , journal=. Time-uniform

  49. [49]

    , author=

    Estimating means of bounded random variables by betting. , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

  50. [50]

    Stochastic Processes and Their Applications , volume=

    Catoni-style confidence sequences for heavy-tailed mean estimation , author=. Stochastic Processes and Their Applications , volume=. 2023 , publisher=

    2023

  51. [51]

    Journal of Theoretical Probability , volume=

    An inequality for tail probabilities of martingales with differences bounded from one side , author=. Journal of Theoretical Probability , volume=. 2003 , publisher=

    2003

  52. [52]

    Pinelis, Iosif , booktitle=. On the