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REVIEW 3 major objections 4 minor 51 references

Probability of worthwhile effect of monotone-response treatments

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Knowing only the two response marginals, the paper computes the exact worst- and best-case probabilities that a monotone treatment effect exceeds a threshold, via two greedy algorithms.

desk verdict The atomic case is a solid contribution, but the advertised continuous extension rests on a broken proof—send back for major revision. read the letter →

arxiv 2607.14414 v1 pith:TRAATQTT submitted 2026-07-15 econ.EM

Benjamin Côté , Ruodu Wang This is my paper · ORCID
classification econ.EM MSC 60E1549Q22
keywords monotonetreatmentresponseworthwhileeffectcounterfactualcausalitydependenceuncertaintyoptimaltransportpartialidentificationstochasticdominanceworst-casebounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

When a treatment and its absence cannot be observed on the same subject, experiments identify only the marginal distributions of the two responses, so the joint distribution — hence the probability P(Y − X > k) that the effect is clinically worthwhile — is not identified. The paper works under the monotone-treatment-response assumption that the treatment never lowers a response (Y ≥ X), and asks for the worst- and best-case values of this probability over all couplings of the two marginals that respect the constraint. Its central result is that for discrete data with equally weighted atoms, both extremes are produced exactly by two greedy matching algorithms, and the same quantities for general discrete and continuous marginals are obtained as limits under atomization and discretization, with optimal couplings converging along with the values. The payoff is a decision-relevant interval for "probability of worthwhile effect", computable from marginal data alone, without choosing a dependence scheme such as independence or comonotonicity. The construction extends to intermediate degrees of treatment, where the additional observed response distributions tighten the bounds.

What carries the argument

The central mechanism is the pair of greedy "least-nuisance" coupling algorithms (Algorithms A and B), which pair atoms of the control distribution with atoms of the treatment distribution in descending order of location, each atom choosing by a fixed preference rule. The admissible objects are semi-couplings H(μ,ν) — plans whose first marginal is exactly μ and whose second marginal is no larger than ν — which exist precisely when μ ≤_st ν; the cost functions are {0,1,∞}-valued indicator functions that enforce y ≥ x. That discrete cost structure is what makes the greedy argument work: any fractional plan can be converted into an atomic one with equal or better objective value. A stability bo

What would settle it

Brute-force enumerate all couplings satisfying y ≥ x for two small atomic distributions (say five atoms each, with μ ≤_st ν) and compare the exact extremes of P(Y − X > k) with the outputs of Algorithms A and B; any mismatch would refute Theorem 1. For the continuous extension, take absolutely continuous marginals whose density is not piecewise Lipschitz (a jump, or unbounded support) and test whether the Section 3.4 discretization values converge as the grid refines — Proposition 4 guarantees nothing in that regime.

Watch

Extended reading notes

Core claim

Central claim (Theorem 1): when μ and ν are atomic with equal atom size and μ ≤_st ν, the smallest and largest possible values of P(Y − X > k) subject to Y ≥ X are achieved exactly by the couplings built by two greedy algorithms. Algorithm A processes μ's atoms from largest to smallest, coupling each to the largest remaining ν-atom within distance k, or else to the largest remaining ν-atom; Algorithm B couples each to the smallest remaining ν-atom with gap exceeding k, or else to the smallest remaining atom with y ≥ x. The proof converts any admissible coupling, right to left, into an atomic one no worse for the objective, so the greedy form is optimal rather than approximate. General discre

Load-bearing premise

The load-bearing premise is that the two response marginal distributions are exactly identified and satisfy μ ≤_st ν, without which no monotone coupling exists; for continuous marginals, the discretization limit is proved only for piecewise Lipschitz densities with ν supported on a compact interval (Section 3.4), and there the best-case value need not be attained, since optimal mass can lie exactly on the boundary y = x + k.

Editorial extensions

If this is right

  • For any marginals satisfying μ ≤_st ν, the probability of a worthwhile effect is confined to the explicit interval between Q_inf_k and Q_sup_k, so a researcher can report a sharp range from marginal data alone rather than guessing a dependence scheme.
  • The optimal couplings are constructed alongside the optimal values, so the dependence structure achieving each extreme is available for inspection; for discrete data the extreme probabilities are exact multiples of the atom size.
  • The stability bound in Theorem 3 means that small perturbations of the marginals move the infimum by no more than the perturbed mass, which justifies computing bounds on atomized or discretized approximations of the true distributions.
  • For threshold k = 0 the best case is the comonotonic coupling, while for k > 0 the extremal couplings are generally neither comonotonic nor counter-monotonic, as the paper exhibits through examples.
  • The same greedy machinery solves the companion dependence-uncertainty bounds on P(X + Y > k) under an ordering constraint, and the adapted algorithms for partial-treatment levels show how additional observed response distributions tighten the interval.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Read as a tool for partial identification, the result means any study that can credibly identify only response marginals can still report sharp bounds on the probability that a treatment is clinically worthwhile — the quantity a clinician or policymaker actually decides on — instead of falling back on the average treatment effect, which cannot answer threshold questions.
  • The non-attainment of the upper bound in the continuous case (mass can sit exactly on y = x + k) suggests that reported best-case probabilities for continuous outcomes should be understood as limiting suprema; using a strict inequality for the event in numerical work would restore attainability.
  • The greedy argument is specialized to {0,1,∞}-valued indicator costs; a natural testable extension would be multi-threshold objectives such as the sum of two crossing probabilities, where per-atom decisions no longer commute and exact greedy optimality may fail.
  • If the marginal densities are not piecewise Lipschitz or the treated response has unbounded support, the paper's discretization convergence proof does not apply, so practitioners should verify convergence numerically before trusting bounds computed for such heavy-tailed or irregular data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies sharp bounds on the probability P(Y−X>k) that a treatment has a clinically worthwhile effect, under the monotone-treatment-response assumption Y≥X and with only the marginal distributions µ (control) and ν (treated) identified. The main result, Theorem 1, gives exact greedy-coupling algorithms (Algorithms A and B) for the worst- and best-case probabilities when µ and ν are atomic with equal atom size. The paper then extends the worst-case result by atomization to general discrete measures (Proposition 3, Theorem 4) and by discretization to absolutely continuous measures with piecewise Lipschitz densities and compact support (Proposition 4, Theorem 5); a parallel sup-problem extension is sketched in Appendix E. Section 4 adds partial-treatment constraints, leading to Algorithms Aη and Bη and a corresponding optimality theorem (Theorem 6). The paper also discusses connections to directional optimal transport and dependence-uncertainty risk aggregation.

Significance. If fully established, the paper would provide an exact, algorithmically implementable solution to a natural partial-identification problem in causal inference, complementing the average-treatment-effect literature. Theorem 1 is a clean, nontrivial result: the greedy algorithms are explicit, the proof in Appendix A is detailed, and the worked examples match the algorithms. This part of the paper is a genuine contribution. The continuous extension is a substantial advertised feature, and it is currently not proven; the proof of Proposition 4 contains a demonstrably false inequality. The paper is honest about its main identifying assumptions (known marginals, stochastic dominance, and regularity conditions for the continuous case), but the central claim of a solved continuous problem is not supported by the present manuscript.

major comments (3)
  1. [Appendix A.8, Proposition 4] The key inequality (S.32) is false. Let µ=U(0,1), ν=U(1,2), k=0.5, t=0, s=1. Definitions (14)-(15) give bµ^0=δ_{0.5} and bν^0=δ_{1.5}, so Q_inf_k(bµ^0,bν^0)=1. The spread measures in (S.28)-(S.29) are bµ_0^1=0.5δ_{0.25}+0.5δ_{0.75} and bν_0^1=0.5δ_{1.25}+0.5δ_{1.75}; the best coupling gives Q_inf=0.5, contradicting the claimed inequality 0.5≥1. Since (S.32) is the bridge from the atomic theorem to the continuous case (through (S.35)-(S.40)), Proposition 4 is not proved. The continuous extension claimed in Section 3.4 and used in Example 8 is therefore currently unsupported.
  2. [Appendix A.8, Theorem 5] Theorem 5 inherits the flaw in Proposition 4: its proof uses the same bound (S.39) and the convergence asserted in (S.40). Thus the existence of the limiting transport plan γ^A_{k,µ,ν} for absolutely continuous marginals is also not established. The analogous sup-problem statements in Appendix E, which rely on the same discretization framework, are likewise left without a valid proof. The load-bearing defect is in the bridge between the atomic/discrete results and the continuous results, not in Theorem 1 itself.
  3. [Appendix A.8, kernel construction] The kernel construction immediately after (S.32) is not well defined: the displayed formula for v_{x,ε} uses an undefined v on both sides of the equation and also has a problematic dependence on the atom mass K^*_s(x,{v}) which may be zero. This makes it impossible to verify the claimed construction of a coupling in H(bµ^(t),bν^(t)). At minimum this passage needs to be rewritten with a clear definition and a rigorous argument that the resulting kernel yields a valid coupling.
minor comments (4)
  1. [Proof of Theorem 2 (Appendix A.3)] There are typos in the weak-convergence argument: 'Since P_2(π_2)≤ν and P_1(π_j)→P_1(π) weakly, then P_1(π)≤ν' should refer to the second marginals throughout. Also, the indexing of π_j and π_2 is inconsistent.
  2. [Appendix A.8, after (S.32)] The formula uses '2s(ε+2^{-(s+1)})' where the context suggests '2^s(...)'. The notation should be corrected to avoid confusion.
  3. [Section 3.4, after (15)] The sentence 'the intervals are designed such they divide in two going from discretization level t to s+1' should be 'from s to s+1'.
  4. [General] The abstract and introduction state that the problems are 'solved' for continuous marginals, but Proposition 4 requires piecewise Lipschitz densities and compact support of ν. The main text should make this limitation prominent in the abstract or introduction, especially if the continuous proof is repaired.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central atomic result is proved from the definitions, and the continuous extension is a convergence claim rather than an input.

full rationale

The paper's central claim, Theorem 1, states that the infimum and supremum in (8)-(9) are attained by Algorithms A and B for atomic measures of the same size. The quantities Q_inf_k and Q_sup_k are defined as extrema over the semi-coupling set H(μ,ν), while Algorithms A and B are independently specified greedy procedures. The proof in Appendix A does not assume the value of Q_inf_k or Q_sup_k; it first reduces to atomic couplings with size a and then proves optimality by induction on the number of atoms, using only the definitions of the feasible set and stochastic dominance. No fitted parameter is later renamed as a prediction, and no target quantity is inserted into the construction. The later discrete and continuous extensions are obtained by atomization (12)-(13) and discretization (14)-(15), with convergence asserted in Propositions 3-4 and Theorems 4-5. Whether Proposition 4's proof is valid (the skeptic's counterexample to inequality (S.32)) is a mathematical correctness concern, not a circularity concern: the continuous value is not assumed as an input, so even a flawed convergence proof would not make the derivation circular. The self-citations appear in auxiliary roles: Chen et al. (2022) and Nutz and Wang (2022) are cited for context, comparison, and background on directional optimal transport, and Côté and Wang (2026) is cited only to extend a standard convex-order fact to infinite means inside the proof of Theorem 2/Proposition 2. These citations do not supply the paper's own conclusions and are not load-bearing for Theorem 1. No uniqueness theorem from the authors' prior work is invoked to forbid alternatives, and no ansatz is smuggled in via citation. Accordingly, there is no circular step to exhibit.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted: k is an input threshold, a is data granularity, and all algorithm outputs are derived from the input marginals. The paper relies on standard measure-theoretic and optimal-transport background, on the domain assumptions of exactly identified marginals and stochastic dominance, and on stated regularity conditions for continuous approximations. No new physical or statistical entities are postulated.

assumptions (6)
  • standard math All random variables live on an atomless probability space (Ω, F, P).
    Stated in Section 2; used throughout to allow standard coupling and limiting arguments.
  • standard math The set M of finite Borel measures on R with stochastic order ≤_st is a lattice, and μ ≤_st ν is equivalent to survival-function dominance and to increasing-test-function comparisons.
    Used in Section 2 to define semi-couplings and in proofs relying on lattice operations and stochastic dominance.
  • standard math H(μ,ν) is nonempty if and only if μ ≤_st ν (Müller and Stoyan, Theorem 2.6.3).
    Invoked after equation (7) to justify feasibility of monotone couplings.
  • domain assumption Monotone treatment response holds almost surely (Y ≥ X), and the marginal distributions μ and ν are fully identified.
    This is the defining scientific assumption of the paper, used in the definitions of Q_inf_k and Q_sup_k and in the admissible coupling set H(μ,ν).
  • domain assumption For the absolutely continuous extension, densities f_μ and f_ν are piecewise Lipschitz continuous and supp(ν) is compact.
    Required in Proposition 4 and Theorem 5 for the discretization to have finite mass and for convergence of Q_inf_k and the optimal couplings.
  • standard math External theorems used in proofs: Villani's Theorem 5.10 for attainment in optimal transport, Rüschendorf's convex-order result for comonotonic/countermonotonic extremality, the Portmanteau theorem, and the Radon–Nikodym theorem.
    Used in the proofs of Theorems 2, 3, and Proposition 2 and in Appendix F.

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Cite this review

Pith. "Pith review of Probability of worthwhile effect of monotone-response treatments." pith.science (2026). https://pith.science/paper/TRAATQTT

@misc{pith2026260714414,
  author       = {Pith},
  title        = {Pith review of: Probability of worthwhile effect of monotone-response treatments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRAATQTT}},
  note         = {Machine review of arXiv:2607.14414}
}
read the original abstract

Experiments may, by design, prevent one from observing on a single subject both the response to a treatment and to its absence. Because of this, marginal distributions for both cases may be observable but not their joint distribution, thus obscuring the distribution of the treatment effect. We examine the case where we impose that the treatment effect is nonnegative, also called monotone treatment response, a common assumption relevant to many practical applications. We solve the problems of best- and worst-case probabilities that the treatment effect exceeds a given value, using an explicit construction for the dependence scheme in each case. Such problems can equivalently be described, in different contexts, as risk aggregation under dependence uncertainty and an order constraint, and as optimal transport with a particular cost function.

Figures

Figures reproduced from arXiv: 2607.14414 by the authors.

Figure 1
Figure 1. Illustration of the coupling multiset ΓA k,µ,ν produced by Algorithm A, for k = 3, for the two discrete distributions with same-size atoms from Example 4. In green are the pairs such that y − x > 3; in red are the ones such that y − x ≤ 3. multiset ΓA 3,µ,ν produced in the process is Γ A 3,µ,ν = {(4.5, 7),(4, 6),(2, 4),(0.5, 3.5),(0, 5.5)}. This coupling multiset is depicted in [PITH_FULL_IMAGE:figures/full_fig_p01… view at source ↗
Figure 2
Figure 2. Illustration of the coupling multiset ΓB k,µ,ν produced by Algorithm B, for k = 2, for the two discrete distributions with same-size atoms from Example 5. In green are the pairs such that y − x > 2; in red are the ones such that y − x ≤ 2. The next result highlights some stability from the algorithms, in the sense that the coupling multiset obtained may be separated in two parts, depending on whether atom pairs meet… view at source ↗
Figure 3
Figure 3. Transport plan γ A 1,µ,ν from Example 7. unbounded measures, but a possible workaround is by finding t ∈ R such that µ|[t,∞) ≤ ν [−k] |[t+k,∞) , where ν [−k] (A) = ν(A + k) for every Borel set A on R. Then, it is certain that all atom locations in [t, ∞) are able to be coupled so that they satisfy the objective y − x ≤ k, specifically by coupling every x to y = x + k. We continue by constructing the coupling leftwar… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Transport plan γ A 1,µ,ν from Example 8. true for every s ∈ N; indeed, for every z ∈ supp(µs ) ∪ supp(νs), µs ([z, ∞)) ≤ µ hz − 2 −(s+1) , ∞  ≤ ν hz − 2 −(s+1) , ∞  ≤ νs ([z, ∞)). Proposition 4. Let µ, ν ∈ M have respective densities fµ and fν, and assume that µ …
Figure 5
Figure 5. Figure 5: The coupling in the left panel is no longer valid if one requires monotonicity of treatment [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Illustration of the coupling multiset produced by Algorithm [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Illustration of the coupling multiset produced by Algorithm [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

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Works this paper leans on

51 extracted references · 3 linked inside Pith

  1. [1]

    R., Fletcher, J

    Amin, V., Behrman, J. R., Fletcher, J. M., Flores, C. A., Flores-Lagunes, A., Kohler, I., Kohler, H.-P. and Stites, S. D. (2025). Causal effects of schooling on memory at older ages in six low-and middle-income countries: Nonparametric evidence with harmonized datasets. Journals of Gerontology, Series B: Psychological Sciences and Social Sciences, 80(6), gbaf057

    2025

  2. [2]

    and Ziegel, J

    Arnold, S., Molchanov, I. and Ziegel, J. F. (2020). Bivariate distributions with ordered marginals. Journal of Multivariate Analysis, 177, 104585

    2020

  3. [3]

    and Pearl, J

    Balke, A. and Pearl, J. (1994). Counterfactual probabilities: computational methods, bounds and applications. Uncertainty in Artificial Intelligence: Proceedings of the 10th Conference, 46--54

    1994

  4. [4]

    and Wang, R

    Bernard, C., Jiang, X. and Wang, R. (2014). Risk aggregation with dependence uncertainty. Insurance: Mathematics and Economics, 54, 93--108

    2014

  5. [5]

    and Wang, R

    Blanchet, J., Lam, H., Liu, Y. and Wang, R. (2025). Convolution bounds on quantile aggregation. Operations Research, 73(5), 2761--2781

    2025

  6. [6]

    M., Cappelen, A

    Bott, K. M., Cappelen, A. W., S rensen, E. . and Tungodden, B. (2020). You’ve got mail: A randomized field experiment on tax evasion. Management Science, 66(7), 2801--2819

    2020

  7. [7]

    and Wang, R

    Chen, Y., Lin, L. and Wang, R. (2022). Risk aggregation under dependence uncertainty and an order constraint. Insurance: Mathematics and Economics, 102, 169--187

    2022

  8. [8]

    Cho, S. J. (2022). The effect of aging out of the Women, Infants, and Children (WIC) program on food insecurity. Health Economics, 31(4), 664--685

    2022

Show all 51 references
  1. [9]

    and Wang, R

    C\^ot\'e, B. and Wang, R. (2026). On convex order and supermodular order without finite mean. Insurance: Mathematics and Economics, 128, 103234

    2026

  2. [10]

    Dawid, A. P. (2000). Causal inference without counterfactuals. Journal of the American Statistical Association, 95(450), 407--424

    2000

  3. [11]

    Dawid, A. P. (2015). Statistical causality from a decision theoretic perspective. Annual Review of Statistics and its Application, 2, 272--303

    2015

  4. [12]

    Demidenko, E. (2016). The p-value you can’t buy. The American Statistician, 70(1), 33--38

    2016

  5. [13]

    and Puccetti, G

    Embrechts, P. and Puccetti, G. (2010). Risk aggregation. In Copula Theory and Its Applications: Proceedings of the Workshop Held in Warsaw, 25-26 September 2009, 111--126

    2010

  6. [14]

    and R\"uschendorf, L

    Embrechts, P., Puccetti, G. and R\"uschendorf, L. (2013). Model uncertainty and VaR aggregation. Journal of Banking and Finance, 37(8), 2750--2764

    2013

  7. [15]

    and Wang, R

    Embrechts, P., Wang, B. and Wang, R. (2015). Aggregation-robustness and model uncertainty of regulatory risk measures. Finance and Stochastics, 19, 763--790

    2015

  8. [16]

    Friesecke, G. (2024). Optimal Transport: A Comprehensive Introduction to Modeling, Analysis, Simulation, Applications. Society for Industrial and Applied Mathematics, Philadelphia

    2024

  9. [17]

    Froehlich, G. W. (1999). What is the chance that this study is clinically significant? A proposal for Q values. Effective Clinical Practice, 2, 234--239

    1999

  10. [18]

    and Schellhorn, M

    Gerfin, M. and Schellhorn, M. (2006). Nonparametric bounds on the effect of deductibles in health care insurance on doctor visits---Swiss evidence. Health Economics, 15(9), 1011--1020

    2006

  11. [19]

    P., Brittain, E

    Greenland, S., Fay, M. P., Brittain, E. H., Shih, J. H., Follmann, D. A., Gabriel, E. E. and Robins, J. M. (2020). On causal inferences for personalized medicine: How hidden causal assumptions led to erroneous causal claims about the D-value. The American Statistician, 74(3), 243--248

    2020

  12. [20]

    and Kreider, B

    Gundersen, C. and Kreider, B. (2009). Bounding the effects of food insecurity on children’s health outcomes. Journal of Health Economics, 28(5), 971--983

    2009

  13. [21]

    and Pepper, J

    Gundersen, C., Kreider, B. and Pepper, J. (2012). The impact of the National School Lunch Program on child health: a nonparametric bounds analysis. Journal of Econometrics, 166(1), 79--91

    2012

  14. [22]

    and Pepper, J

    Gundersen, C., Kreider, B. and Pepper, J. V. (2017). Partial identification methods for evaluating food assistance programs: a case study of the causal impact of SNAP on food insecurity. American Journal of Agricultural Economics, 99(4), 875--893

    2017

  15. [23]

    Hand, D. J. (1992). On comparing two treatments. The American Statistician, 46(3), 190--192

    1992

  16. [24]

    Hof, S. (2014). Does private tutoring work? The effectiveness of private tutoring: a nonparametric bounds analysis. Education Economics, 22(4), 347--366

    2014

  17. [25]

    and Sturm, K.-T

    Huesmann, M. and Sturm, K.-T. (2013). Optimal transport from Lebesgue to Poisson. Annals of Probability, 41(4), 2426--2478

    2013

  18. [26]

    Q., and Raban, D

    Jaffe, A. Q., and Raban, D. (2025). Coupling theory, optimal transport, and Strassen's theorem beyond regular orders. arXiv:2509.21616

    2025 arXiv

  19. [27]

    and Spector, A

    Ji, W., Lei, L. and Spector, A. (2023). Model-agnostic covariate-assisted inference on partially identified causal effects. arXiv:2310.08115

    2023 arXiv

  20. [28]

    Kim, J. H. (2014). Identifying the distribution of treatment effects under support restrictions. arXiv:1410.5885

    2014 arXiv

  21. [29]

    Kreider, B., Pepper, J. V. and Roy, M. (2016). Identifying the effects of WIC on food insecurity among infants and children. Southern Economic Journal, 82(4), 1106--1122

    2016

  22. [30]

    and Wang, R

    Liu, F. and Wang, R. (2021). A theory for measures of tail risk. Mathematics of Operations Research, 46(3), 1109--1128

    2021

  23. [31]

    B., and Whitney, D

    Mann, H. B., and Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics, 18(1), 50--60

    1947

  24. [32]

    Manski, C. F. (1994). The selection problem. Advances in Econometrics, 1, 147--170. Cambridge University Press, Cambridge

    1994

  25. [33]

    Manski, C. F. (1995). Identification Problems in the Social Sciences. Harvard University Press, Cambridge

    1995

  26. [34]

    Manski, C. F. (1997). Monotone treatment response. Econometrica, 65(6), 1311--1334

    1997

  27. [35]

    Manski, C. F. (2007). Identification for Prediction and Decision. Harvard University Press, Cambridge

    2007

  28. [36]

    Manski, C. F. and Pepper, J. V. (2000). Monotone instrumental variables, with an application to the returns to schooling. Econometrica, 68(4), 997--1010

    2000

  29. [37]

    Manski, C. F. and Pepper, J. V. (2009). More on monotone instrumental variables. Econometrics Journal, 12(1), S200--S216

    2009

  30. [38]

    and Wang, R

    Mao, T. and Wang, R. (2015). On aggregation sets and lower-convex sets. Journal of Multivariate Analysis, 138, 170--181

    2015

  31. [39]

    and Stoyan, D

    M\"uller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley, Hoboken

    2002

  32. [40]

    and Wang, R

    Nutz, M. and Wang, R. (2022). The directional optimal transport. Annals of Applied Probability, 32(2), 1400--1420

    2022

  33. [41]

    Rachev, S. T. and R \"u schendorf, L. (2006). Mass Transportation Problems, Volume 1: Theory. Springer, New York

    2006

  34. [42]

    R\"uschendorf, L. (1982). Random variables with maximum sums. Advances in Applied Probability, 14(3), 623--632

    1982

  35. [43]

    R \"u schendorf, L. (2013). Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios . Springer, Heidelberg

    2013

  36. [44]

    Senn, S. (2009). Three things that every medical writer should know about statistics. Journal of the European Medical Writers Association, 18(3), 159--162

    2009

  37. [45]

    and Shanthikumar, J

    Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York

    2007

  38. [46]

    Villani, C. (2009). Optimal Transport: Old and New. Springer, Heidelberg

    2009

  39. [47]

    and Wu, Q

    Wang, R. and Wu, Q. (2025). The reference interval in higher-order stochastic dominance. Economic Theory Bulletin, 13, 263--277

    2025

  40. [48]

    and Zitikis, R

    Wang, R. and Zitikis, R. (2021). An axiomatic foundation for the Expected Shortfall. Management Science, 67(3), 1413--1429

    2021

  41. [49]

    Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80-83

    1945

  42. [50]

    and Morgan, S

    Winship, C. and Morgan, S. L. (1999). The estimation of causal effects from observational data. Annual Review of Sociology, 25(1), 659--706

    1999

  43. [51]

    Yarnell, C. J. and Goligher, E. C. (2024). Interpreting posterior probabilities in Bayesian analyses of clinical trials. Lancet Respiratory Medicine, 12(3), 188--190

    2024

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