# Bentkus-type asymptotic e-values

arXiv: https://arxiv.org/abs/2606.06332 · HTML: https://mlsanity.com/paper/2606.06332/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 4 Jun 2026.

## Builds on (most cited first, 22 of 22)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Hypothesis testing with e-values](https://mlsanity.com/paper/2410.23614/) (2024)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [E-Values Expand the Scope of Conformal Prediction](https://mlsanity.com/paper/2503.13050/) (2025)
- [Catoni-style confidence sequences for heavy-tailed mean estimation](https://mlsanity.com/paper/2202.01250/) (2022)
- [Post-selection inference for e-value based confidence intervals](https://mlsanity.com/paper/2203.12572/) (2022)
- [Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha](https://mlsanity.com/paper/2205.00901/) (2022)
- [Principal-Agent Hypothesis Testing](https://mlsanity.com/paper/2205.06812/) (2022)
- [Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence](https://mlsanity.com/paper/2411.11271/) (2024)
- [Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity](https://mlsanity.com/paper/2511.03606/) (2025)
- [Post-Hoc Large-Sample Statistical Inference](https://mlsanity.com/paper/2603.08002/) (2026)
- [Admissible online closed testing must employ e-values](https://mlsanity.com/paper/2407.15733/) (2024)
- [Asymptotic and compound e-values: multiple testing and empirical Bayes](https://mlsanity.com/paper/2409.19812/) (2024)
- [On admissibility in post-hoc hypothesis testing](https://mlsanity.com/paper/2508.00770/) (2025)
- [Rao-Blackwellized e-variables](https://mlsanity.com/paper/2512.16759/) (2025)
- [An optimal three-way stable and monotonic spectrum of bounds on quantiles: a spectrum of coherent measures of financial risk and economic inequality](https://mlsanity.com/paper/1310.6025/) (2013)
- [Optimal E-Values for Exponential Families: the Simple Case](https://mlsanity.com/paper/2404.19465/) (2024)
- [Bringing Closure to False Discovery Rate Control: A General Principle for Multiple Testing](https://mlsanity.com/paper/2509.02517/) (2025)
- [Equivalence testing with data-dependent and post-hoc equivalence margins](https://mlsanity.com/paper/2603.16213/) (2026)
- [E-Values for Exponential Families: the General Case](https://mlsanity.com/paper/2409.11134/) (2024)

## Abstract (arXiv)

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.