arxiv: 2604.21851 · v1 · submitted 2026-04-23 · 📊 stat.ME · math.ST· stat.TH

Recognition: unknown

Betting on Bets: Anytime-Valid Tests for Stochastic Dominance

Ilia Tsetlin, Marco Scarsini, Sebastian Arnold, Yo Joong Choe

Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:53 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH

keywords stochastic dominanceanytime-valid testse-processessequential testingnonparametric inferencestochastic orderingpower one tests

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The pith

Sequential tests for stochastic dominance achieve power one by mixing asymptotically growth-rate optimal e-variables.

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

The paper develops new sequential tests that let researchers monitor in real time whether one uncertain prospect is better than another in a distributional sense, beyond just averages. These tests accumulate evidence against the idea that one is dominated by the other using a new construction based on betting-like quantities that grow optimally. The resulting procedures will eventually detect dominance if it exists and stay valid no matter when you look at the data. This works not only for the basic version but also for stronger orders of dominance. They perform as well as traditional fixed-sample tests but without needing to decide the sample size in advance.

Core claim

We develop a novel family of sequential, anytime-valid tests for stochastic dominance by constructing e-processes as mixtures of asymptotically growth-rate optimal e-variables. These yield power-one tests that remain valid under continuous monitoring. The construction is nonparametric for first-order stochastic dominance and extends directly to any higher-order version.

What carries the argument

Mixture of asymptotically growth-rate optimal e-variables forming an e-process that quantifies accumulating evidence against the null of no stochastic dominance.

If this is right

The tests achieve power one, eventually rejecting the null almost surely when dominance holds.
Validity is preserved for any stopping time or continuous monitoring schedule.
The same mixing construction applies to higher-order stochastic dominance.
Empirical power matches or approaches that of standard non-sequential tests for fixed samples.

Where Pith is reading between the lines

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

The framework could support ongoing monitoring of investment or treatment prospects without pre-fixing sample sizes.
Conditions sketched for testing the complementary non-dominance null might enable detection of definite upside in sequential settings.
Similar mixtures could be explored for related ordering concepts such as convex stochastic order.

Load-bearing premise

Mixtures of asymptotically growth-rate optimal e-variables for the stochastic dominance null will produce processes that grow without bound under the alternative while staying valid at all times.

What would settle it

A data-generating process where one distribution truly dominates the other yet the constructed evidence process remains bounded, or where the test rejects the null too often when monitoring continuously on identical distributions.

Figures

Figures reproduced from arXiv: 2604.21851 by Ilia Tsetlin, Marco Scarsini, Sebastian Arnold, Yo Joong Choe.

**Figure 1.** Figure 1: All GRO e-processes grow quickly under anti-monotonicity while maintaining anytime-validity. Plots show simulations with finite support (4 outcomes) and maximal anti-monotonicity (𝜌(𝑋, 𝑌) = −1). Each line is averaged over 500 repeated simulations. The Ville error plot is drawn for 𝑡 = 1, … , 5, 000, while the e-power plot is truncated at 𝑡 = 500 and at value 40 for better visualization. Over time, a linear… view at source ↗

**Figure 2.** Figure 2: Adaptive GRO e-processes can grow even when the non-dominance region is far away from the initial search interval (Case 4 vs. Case 3). They can also grow more quickly than non-adaptive GRO eprocesses when the contact set is relatively small within the initial search interval (Case 2). All methods control the Ville error at level 𝛼 = 0.05 under the “hardest” null case (Case 1, where 𝐹𝑋 ≡ 𝐹𝑌). The e-power p… view at source ↗

**Figure 3.** Figure 3: All GRO e-processes grow quickly even when the contact set is large, with the adaptive variant with exponential weights having the largest e-power. These plots summarize simulations where 𝐹𝑌 ≡ 𝖴𝗇𝗂𝖿[0, 1] and 𝐹𝑋 is a piecewise uniform distribution (25) with a kink point at 𝑧0 ∈ [0, 1]. As 𝑧0 decreases, the two CDFs have more “contact” and the testing problem becomes more challenging (under the null, to not… view at source ↗

**Figure 4.** Figure 4: When compared against classical, non-anytime-valid approaches, GRO e-processes can achieve competitive (if not better) power. We compare the level-𝛼 sequential tests induced by adaptive and nonadaptive GRO e-processes (AdaGRO-Exp and GRO, respectively) against three classical, non-anytimevalid test (BD03, LMW05, and LSW10). The level 𝛼 is fixed at 0.05. Figure 4a confirms that, under the null (𝑧0 = 0, i.… view at source ↗

**Figure 5.** Figure 5: For higher-order SD testing, adaptive UP e-processes grow quickly, relative to non-adaptive UP or constant-bet counterparts, particularly when the contact set between the CDFs is large. These plots summarize simulations in the same setup as for [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

**Figure 6.** Figure 6: Histograms and means of the stopping times [PITH_FULL_IMAGE:figures/full_fig_p053_6.png] view at source ↗

**Figure 7.** Figure 7: All UP e-processes control the Ville error at level [PITH_FULL_IMAGE:figures/full_fig_p054_7.png] view at source ↗

read the original abstract

How can we monitor, in real time, whether one uncertain prospect has any upside over another? To answer this question, we develop a novel family of sequential, anytime-valid tests for stochastic dominance (SD; also known as stochastic ordering), a classical and popular notion for comparing entire distribution functions. The problem is distinct from the popular problem of testing for dominance in means, which would not capture distributional differences beyond the first moment. We first derive powerful, nonparametric e-processes that quantify evidence against the null hypothesis that one prospect is dominated by another. For first-order SD, these e-processes are constructed as a mixture of asymptotically growth-rate optimal e-variables and yield a test of power one. The approach further generalizes to sequential testing for SD beyond the first order, including any higher-order SD. Empirically, we demonstrate that the resulting sequential tests are competitive with existing non-sequential SD tests in terms of power, while achieving validity under continuous monitoring that existing methods do not. Finally, we sketch the complementary and challenging problem of testing the non-SD null hypothesis, which asks whether a prospect has a definite upside, and describe the conditions under which we can derive a nontrivial anytime-valid test.

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.

Desk Editor's Note private letter to a colleague

The paper builds anytime-valid tests for stochastic dominance by mixing asymptotically optimal e-variables, delivering power-one guarantees under continuous monitoring.

read the letter

The main advance here is a family of e-processes for testing stochastic dominance that stay valid no matter when you look at the data. They mix growth-rate optimal e-variables to get power one under the alternative, and the same idea extends to higher-order dominance. That is genuinely new for this nonparametric setting, where most existing tests are fixed-sample only. The empirical comparisons show the sequential versions keep up with standard SD tests in power while adding the anytime-valid property, which matters for real-time monitoring in finance or online experiments. They also sketch how one might approach the harder problem of testing the non-dominance null, though that part stays preliminary. The construction follows the established e-process playbook, so the validity claims look internally consistent on paper. The soft spots are practical rather than foundational. The power-one result depends on the existence and quality of those base optimal e-variables for the composite SD null; if the mixing measure is misspecified or the distributions are close, finite-sample behavior could degrade faster than the asymptotics suggest. The experiments are mostly synthetic, so it is not yet clear how the tests perform on messy real data with ties or heavy tails. Overall this is aimed at researchers in sequential analysis and decision theory who already work with e-processes or need distribution-free monitoring tools. It is not a broad shift but fills a specific gap cleanly enough to deserve referee time. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The paper develops a family of sequential, anytime-valid tests for stochastic dominance (SD) using nonparametric e-processes. For first-order SD, the e-processes are constructed as mixtures of asymptotically growth-rate optimal e-variables and are claimed to yield power-one tests under the alternative while remaining valid under continuous monitoring. The approach generalizes to higher-order SD, and the manuscript includes empirical comparisons showing competitiveness with non-sequential SD tests plus a sketch of the complementary problem of testing the non-SD null.

Significance. If the derivations and power-one claims hold, the work supplies a practical tool for real-time monitoring of distributional dominance that existing fixed-sample tests lack. The grounding in e-process theory for a composite nonparametric null is a clear strength, as is the explicit generalization beyond first-order SD and the empirical demonstration of power competitiveness. These features address a genuine gap in sequential nonparametric testing with direct relevance to economics and decision theory.

major comments (2)

[§3.2] §3.2, the mixture construction: the claim that the e-process is a mixture of asymptotically growth-rate optimal e-variables for the first-order SD null requires an explicit statement of the mixing measure and a proof that the resulting process retains the growth-rate optimality (or at least the power-one property) under the alternative; without this, the power-one guarantee cannot be verified from the given construction.
[§4] §4, generalization to higher-order SD: the extension from first-order to k-th order SD is sketched but the corresponding e-variable family and the validity argument under continuous monitoring are not derived in detail; this is load-bearing for the claim that the method 'further generalizes' and needs at least a theorem statement with the key steps.

minor comments (2)

[Throughout] Notation for the e-processes (e.g., the distinction between the instantaneous e-variable and the cumulative process) is introduced without a consolidated table or definition list, making it easy to lose track across sections.
[§5] The empirical section would benefit from reporting the exact sample sizes and number of Monte Carlo replications used to generate the power curves, as well as a direct comparison of type-I error under continuous monitoring versus fixed-sample benchmarks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation for minor revision. We address the two major comments point by point below, agreeing that greater explicitness is needed in both cases. The requested clarifications will be incorporated into the revised manuscript.

read point-by-point responses

Referee: [§3.2] §3.2, the mixture construction: the claim that the e-process is a mixture of asymptotically growth-rate optimal e-variables for the first-order SD null requires an explicit statement of the mixing measure and a proof that the resulting process retains the growth-rate optimality (or at least the power-one property) under the alternative; without this, the power-one guarantee cannot be verified from the given construction.

Authors: We agree that an explicit statement of the mixing measure and a supporting argument for the power-one property are required for full verifiability. In the revision we will state the mixing measure precisely (a probability measure supported on the class of asymptotically growth-rate optimal e-variables for the first-order SD null) and add a short lemma establishing that the resulting mixture e-process inherits the power-one property under the alternative. This addition will be placed in §3.2 immediately after the construction. revision: yes
Referee: [§4] §4, generalization to higher-order SD: the extension from first-order to k-th order SD is sketched but the corresponding e-variable family and the validity argument under continuous monitoring are not derived in detail; this is load-bearing for the claim that the method 'further generalizes' and needs at least a theorem statement with the key steps.

Authors: We acknowledge that the higher-order extension is currently only sketched. We will expand §4 with a formal theorem that (i) defines the family of e-variables for k-th order SD, (ii) states the corresponding e-process, and (iii) outlines the key steps establishing anytime-validity under continuous monitoring. The proof will adapt the first-order argument via the appropriate integral representation of higher-order dominance. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs e-processes for stochastic dominance testing by mixing asymptotically growth-rate optimal e-variables drawn from established sequential testing theory. This step relies on external optimality results for e-variables under composite nonparametric nulls rather than defining the target quantity in terms of itself or fitting parameters to the same data used for validation. No load-bearing self-citations reduce the central claim to unverified prior work by the same authors; the power-one property and continuous-monitoring validity follow directly from the mixture construction and standard e-process martingale properties. The derivation remains self-contained against external benchmarks in e-process literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract; the approach builds on existing e-process theory without introducing new postulated entities.

pith-pipeline@v0.9.0 · 5521 in / 1058 out tokens · 40217 ms · 2026-05-09T20:53:50.734725+00:00 · methodology

discussion (0)

Reference graph

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