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arxiv: 2604.20612 · v1 · submitted 2026-04-22 · 🧮 math.ST · math.PR· stat.TH

E-values and sequential power-one tests for monotonicity and unimodality

Hongjian Wang , Aaditya Ramdas This is my paper

Pith reviewed 2026-05-09 22:51 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH
keywords e-valuese-processesmonotonicity testingunimodalitypower-one testssequential hypothesis testingmode estimationshape-constrained inference
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The pith

E-processes yield power-one sequential tests for monotonicity and unimodality along with consistent mode estimators.

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

The paper develops e-values and e-processes to test whether a distribution over nonnegative integers is monotone or whether a distribution over integers is unimodal at a specified mode. These constructions support sequential tests that reject any false null hypothesis with probability one as more i.i.d. observations arrive. They also produce set-valued estimators for the mode that stabilize exactly at the true mode set after finite time. The work fully characterizes every valid e-value obtainable from a single observation and identifies the most powerful e-value against any fixed alternative. The same approach extends to continuous random variables and aligns with existing techniques in shape-constrained inference.

Core claim

We develop e-values and e-processes testing the null hypothesis that a distribution over nonnegative integers is monotone, and that a distribution over integers is unimodal given a certain mode. Our e-processes lead to tests of power one under any non-null distribution with a sequence of i.i.d. observations, and consistent set-valued mode estimators that eventually equal the true set of modes. Additionally, we characterize the set of all e-values, and therefore the set of all valid tests, with one monotone and unimodal observation, as well as the most powerful e-value for a fixed alternative. We then show that many of our results can be generalized to continuous random variables, relating to

What carries the argument

e-processes for the monotonicity and unimodality hypotheses, built to accumulate evidence sequentially from i.i.d. observations and to produce anytime-valid tests plus consistent mode-set estimators.

If this is right

  • Any false null of monotonicity or unimodality is rejected with probability one after sufficiently many i.i.d. observations.
  • The set-valued mode estimator equals the true mode set after finite time almost surely.
  • Every valid test based on a single observation is described by the complete characterization of e-values.
  • The most powerful e-value against any chosen alternative is explicitly identified.
  • The discrete constructions extend directly to continuous random variables and connect to prior shape-constrained methods.

Where Pith is reading between the lines

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

  • The power-one property could support real-time monitoring of distributional shape in streaming data without pre-specified sample sizes.
  • The characterization of all single-observation e-values supplies a complete menu of valid tests for practitioners to select from.
  • Generalization to continuous cases suggests the framework may unify discrete and continuous shape-constrained testing under one sequential lens.

Load-bearing premise

The observations are independent and identically distributed draws from the unknown distribution, and the null hypothesis is exactly that the distribution is monotone or unimodal at the given mode.

What would settle it

Draw i.i.d. samples from a clearly non-monotone distribution such as a discrete uniform with an artificial dip and verify whether the e-process remains bounded or diverges to infinity with probability one.

read the original abstract

We develop e-values and e-processes testing the null hypothesis that a distribution over nonnegative integers is monotone, and that a distribution over integers is unimodal given a certain mode. Our e-processes lead to tests of power one under any non-null distribution with a sequence of i.i.d. observations, and consistent set-valued mode estimators that eventually equal the true set of modes. Additionally, we characterize the set of all e-values, and therefore the set of all valid tests, with one monotone and unimodal observation, as well as the most powerful e-value for a fixed alternative. We then show that many of our results can be generalized to continuous random variables, relating them to the existing results in the shape-constrained inference literature.

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

2 major / 2 minor

Summary. The paper develops e-values and e-processes for testing the composite null that a distribution on nonnegative integers is monotone and that a distribution on integers is unimodal at a specified mode. It shows that the resulting e-processes are supermartingales under the null, yield power-one tests that reject almost surely under any alternative with i.i.d. observations, and produce consistent set-valued estimators for the mode set. The authors fully characterize the set of all valid e-values for a single observation, identify the most powerful e-value against a fixed alternative, and extend the constructions and properties to continuous distributions while relating them to the shape-constrained inference literature.

Significance. If the constructions and proofs hold, the work supplies a systematic e-value framework for sequential testing of shape constraints, delivering power-one tests and eventual exact recovery of the mode set under minimal i.i.d. assumptions. The complete characterization of single-observation e-values is a notable strength, as it delineates the entire class of valid tests rather than exhibiting isolated examples. The continuous extension connects the discrete results to existing shape-constrained methods, potentially enabling new applications in nonparametric statistics and sequential analysis.

major comments (2)
  1. [Section on e-process construction and validity] The central claim that the constructed e-processes are supermartingales under the composite nulls of monotonicity and unimodality (and diverge a.s. under every alternative) is load-bearing for the power-one and consistency results; the explicit verification of the supermartingale property for the full composite null should be presented with the construction, including boundary cases for the mode.
  2. [Section on characterization of e-values for one observation] The characterization of the full set of e-values for a single observation is used to identify the most powerful e-value for a fixed alternative; this step requires a complete proof that the enumerated set is exhaustive under the null, as any omission would affect the optimality claim.
minor comments (2)
  1. The abstract states the i.i.d. assumption explicitly but the main text should clarify whether the results extend immediately to non-i.i.d. settings or require additional conditions.
  2. Notation for the set-valued mode estimator and its convergence should be introduced with a brief definition early in the paper to improve readability for readers unfamiliar with e-process literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation, and constructive suggestions. We address the two major comments below and will revise the manuscript to incorporate the requested clarifications and expanded proofs.

read point-by-point responses

  1. Referee: [Section on e-process construction and validity] The central claim that the constructed e-processes are supermartingales under the composite nulls of monotonicity and unimodality (and diverge a.s. under every alternative) is load-bearing for the power-one and consistency results; the explicit verification of the supermartingale property for the full composite null should be presented with the construction, including boundary cases for the mode.

    Authors: We agree that an explicit, self-contained verification of the supermartingale property under the full composite null, including all boundary cases for the mode, will improve readability and rigor. In the revised version we will add a dedicated paragraph immediately after the e-process construction that verifies the supermartingale property for every distribution in the composite null (covering interior and boundary modes) and recalls the almost-sure divergence argument under alternatives. revision: yes

  2. Referee: [Section on characterization of e-values for one observation] The characterization of the full set of e-values for a single observation is used to identify the most powerful e-value for a fixed alternative; this step requires a complete proof that the enumerated set is exhaustive under the null, as any omission would affect the optimality claim.

    Authors: We concur that the optimality claim rests on a fully rigorous demonstration that the enumerated collection exhausts all valid e-values. Our current manuscript contains an outline of the argument; we will expand this into a complete, self-contained proof in the revision, detailing each step that shows no other e-value exists outside the characterized set under the null. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives e-values for monotonicity and unimodality hypotheses directly from the definition of an e-value (nonnegative random variable with expectation ≤1 under the null) and extends them to e-processes via the supermartingale property under i.i.d. sampling. The characterization of all valid e-values for a single observation is obtained by explicit construction satisfying the expectation constraint, and power-one consistency follows from standard strong-law arguments on the log of the e-process. These steps are self-contained probabilistic derivations that do not reduce to fitted parameters, self-definitional loops, or load-bearing self-citations; prior e-value results are used only as background and the shape-constrained constructions are independent.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; e-values and e-processes are referenced as developed here but likely extend prior work by the authors.

axioms (2)
  • domain assumption Observations are i.i.d. from the unknown distribution
    Standard assumption stated for the sequential tests and power-one property.
  • domain assumption Null hypothesis is that the distribution is exactly monotone or unimodal at the specified mode
    Core definition of the testing problem in the abstract.

pith-pipeline@v0.9.0 · 5420 in / 1286 out tokens · 68972 ms · 2026-05-09T22:51:59.882438+00:00 · methodology

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Reference graph

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