The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent
Abstract
In safe hypothesis testing with test supermartingales, 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 behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate 𝔼P1[log E], the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level c>0, every α and horizon t we construct e-variables of conditional e-power exactly c whose probability of not rejecting by t is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, Λ(E)=sups≥0{−log 𝔼P1[E−s]}, whose range is exactly determined: supE Λ(E)=KL(P0|P1), the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, KL(P1|P0), and that one is attained, P0-a.s. uniquely, by the likelihood ratio R. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family Rβ/𝔼P0[Rβ]: the ceiling as β↓0, R at β=1. Which β is best is settled by the horizon, exactly: R is optimal at t=log(1/α)/KL(P1|P0) alone, beaten by sharpening (β>1) below it and by flattening above.
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BibTeX
@misc{forre2026typeii,
title = {{The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent}},
author = {Patrick Forré},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27765},
eprint = {2609.27765},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27765},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Type-II Error of Test Supermartingales: e-Power versus the Chernoff-Stein Exponent
AU - Patrick Forré
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27765
UR - https://arxiv.org/abs/2609.27765
AB - In safe hypothesis testing with test supermartingales, 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 behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate 𝔼_P_1[log E], the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level c>0, every α and horizon t we construct e-variables of conditional e-power exactly c whose probability of not rejecting by t is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, Λ(E)=sup_s≥0{−log 𝔼_P_1[E^−s]}, whose range is exactly determined: sup_E Λ(E)=KL(P_0|P_1), the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, KL(P_1|P_0), and that one is attained, P_0-a.s. uniquely, by the likelihood ratio R. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family R^β/𝔼_P_0[R^β]: the ceiling as β↓0, R at β=1. Which β is best is settled by the horizon, exactly: R is optimal at t=log(1/α)/KL(P_1|P_0) alone, beaten by sharpening (β>1) below it and by flattening above.
ER -