Type-II Error Bounds for Test Supermartingales from Lower-Tail Hypotheses
Abstract
In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level α∈(0,1], if one rejects the null hypothesis whenever the wealth process first exceeds 1/α. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level α, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds log1/α. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.
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BibTeX
@misc{forre2026typeii,
title = {{Type-II Error Bounds for Test Supermartingales from Lower-Tail Hypotheses}},
author = {Patrick Forré},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27766},
eprint = {2609.27766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27766},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Type-II Error Bounds for Test Supermartingales from Lower-Tail Hypotheses
AU - Patrick Forré
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27766
UR - https://arxiv.org/abs/2609.27766
AB - In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level α∈(0,1], if one rejects the null hypothesis whenever the wealth process first exceeds 1/α. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level α, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds log1/α. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.
ER -