pith. sign in

arxiv: 2606.22008 · v1 · pith:7Z7KBE5Hnew · submitted 2026-06-20 · 📊 stat.ME · math.PR· math.ST· stat.TH

An Optimal Transportation Approach for Improved Confidence Intervals

Christophe Quentin Valvason , Eustasio del Barrio , Stefan Sperlich This is my paper

Pith reviewed 2026-06-26 11:44 UTC · model grok-4.3

classification 📊 stat.ME math.PRmath.STstat.TH
keywords optimal transportconfidence intervalscoverage probabilityquantile methodsoptimal couplingsstatistical inferencesmall sample performance
0
0 comments X

The pith

Optimal transport can construct confidence intervals with improved coverage by minimizing deviations through optimal couplings.

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

The paper introduces a procedure that applies optimal transport to extend classical quantile-based confidence intervals. It derives a theoretical framework establishing consistency and error bounds for the coverage probability of the resulting intervals. Data-driven choices for hyper-parameters are proposed to ensure practical feasibility. Standard asymptotic methods often perform poorly for small samples or complex problems, so this approach aims to reduce coverage errors where traditional tools fall short.

Core claim

The paper claims that leveraging optimal couplings from optimal transport extends quantile-based confidence intervals in a way that minimizes coverage deviations, while maintaining consistency and controlled error bounds for the coverage probability, with practical hyper-parameter choices enabling its use.

What carries the argument

Optimal couplings chosen to minimize coverage deviations when constructing confidence intervals from empirical measures.

If this is right

  • The new intervals achieve better accuracy and robustness than standard methods in simulations across estimation problems.
  • Consistency of coverage probability holds under the derived theoretical framework.
  • Error bounds quantify the reliability of coverage for the adjusted intervals.
  • Data-driven hyper-parameter selection maintains feasibility while respecting the theoretical guarantees.

Where Pith is reading between the lines

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

  • The same coupling idea could be tested on prediction intervals or tolerance regions where coverage control is also difficult.
  • Links may exist to other distribution-comparison techniques in statistics that rely on geometric distances between measures.
  • Application to high-dimensional or nonparametric settings where quantile methods degrade would provide a direct test of practical value.

Load-bearing premise

That data-driven choices for the hyper-parameters preserve the theoretical consistency and error bounds without introducing new coverage distortions.

What would settle it

Repeated simulations in which the empirical coverage of the new intervals deviates from the nominal level by more than the derived error bound, or fails to improve on classical quantile intervals, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.22008 by Christophe Quentin Valvason, Eustasio del Barrio, Stefan Sperlich.

Figure 1.1
Figure 1.1. Figure 1.1: Average OT-based confidence interval endpoints (for the mean of log-normal [PITH_FULL_IMAGE:figures/full_fig_p003_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Left: Barplot of Q with the admissible set C in blue. Dashed arrows indicate the contribution of atoms in C to the confidence interval In,Q. Right: Density of Pn with the orange-shaded region indicating the confidence interval In,Q. Remark 2.2. Our methodology can be extended to an estimator ˆθn with bias Bn by defining a modified error ϑ (bias) n :“ rn ` ˆθn ´ θ0 ´ Bn ˘ , and repeat the construction wit… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Estimated density of the galaxies velocity dataset. [PITH_FULL_IMAGE:figures/full_fig_p026_5_1.png] view at source ↗
read the original abstract

Optimal transport methods have recently attracted a lot of attention in statistics. Their appeal lies in providing a geometric framework for comparing probability measures, leading to new perspectives on classical problems. A central problem in statistics is the construction of valid confidence sets as fundamental inferential tools in practice. A well-known problem is that for complex problems or relatively small samples, their asymptotic approximations often show poor performance. This suggests to apply optimal transport methods when constructing confidence sets for hard problems to improve their coverage properties. We introduce such a procedure, derive the theoretical framework studying consistency and error bounds for the coverage probability of the resulting intervals. To guarantee feasibility in practice, we propose data-driven choices for our hyper parameters. This approach extends classical quantile-based confidence intervals by leveraging optimal couplings to minimize coverage deviations. Simulations demonstrate striking performance in different estimation problems, outperforming standard methods in accuracy and robustness.

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 / 1 minor

Summary. The paper proposes an optimal transport-based procedure to construct confidence intervals that improve upon classical quantile-based methods by using optimal couplings to minimize coverage deviations from the nominal level. It claims to derive a theoretical framework establishing consistency and error bounds for the coverage probability of the resulting intervals, introduces data-driven choices for hyper-parameters to ensure practical feasibility, and reports simulation results showing superior accuracy and robustness across different estimation problems.

Significance. If the consistency and error-bound results extend to the data-driven hyper-parameter setting, the work could provide a geometrically motivated framework for addressing poor finite-sample coverage in complex or small-sample inference problems where standard asymptotic approximations are unreliable. The explicit focus on coverage error bounds and the simulation comparisons are potential strengths if the theoretical derivations are rigorous and the empirical gains are shown to be robust.

major comments (2)
  1. [theoretical framework / abstract] Abstract and theoretical framework section: The error bounds and consistency results are derived for the optimal coupling procedure, but the manuscript does not supply an argument showing that these bounds remain valid when hyper-parameters are selected in a data-dependent manner (as proposed for practical feasibility). Data-driven selection introduces additional dependence between the coupling and the sample that is not present for fixed hyper-parameters, and it is unclear whether the extra randomness is controlled by the stated coverage error term.
  2. [error bounds derivation] Section deriving error bounds: The central coverage guarantee appears to rely on properties of the optimal coupling for fixed hyper-parameters; no explicit extension or perturbation analysis is provided to cover the case where hyper-parameters are estimated from the same data, which could inflate the coverage deviation beyond the derived bound.
minor comments (1)
  1. [abstract] The abstract refers to 'striking performance' in simulations without specifying the estimation problems, sample sizes, or number of replications; adding these details would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the potential of the optimal transport framework for confidence interval construction. The comments correctly identify a gap in the current theoretical development. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [theoretical framework / abstract] Abstract and theoretical framework section: The error bounds and consistency results are derived for the optimal coupling procedure, but the manuscript does not supply an argument showing that these bounds remain valid when hyper-parameters are selected in a data-dependent manner (as proposed for practical feasibility). Data-driven selection introduces additional dependence between the coupling and the sample that is not present for fixed hyper-parameters, and it is unclear whether the extra randomness is controlled by the stated coverage error term.

    Authors: We agree that the existing consistency and error-bound derivations assume fixed hyper-parameters and do not explicitly treat the data-dependent case. The additional randomness induced by data-driven selection is not controlled in the current proofs. In the revised manuscript we will add a dedicated subsection that supplies a perturbation argument: under the assumption that the data-driven hyper-parameter estimator converges in probability to its population counterpart at rate o_p(1) (which holds for standard cross-validation or plug-in estimators under mild regularity), the coverage deviation remains bounded by the original term plus an additive remainder that vanishes asymptotically. This will be stated as a corollary to the fixed-hyper-parameter result. revision: yes

  2. Referee: [error bounds derivation] Section deriving error bounds: The central coverage guarantee appears to rely on properties of the optimal coupling for fixed hyper-parameters; no explicit extension or perturbation analysis is provided to cover the case where hyper-parameters are estimated from the same data, which could inflate the coverage deviation beyond the derived bound.

    Authors: The referee is correct that the derivation in the error-bounds section is stated only for fixed hyper-parameters and contains no perturbation analysis for the estimated case. We will revise that section to include the required extension, showing that the extra dependence can be absorbed into the existing error term when the hyper-parameter estimator is consistent at a suitable rate. The revised bound will be presented explicitly, together with the conditions under which it holds. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation of coverage bounds remains independent of data-driven hyper-parameter proposal

full rationale

The provided abstract and context describe a procedure that extends quantile-based intervals via optimal couplings, followed by a separate derivation of consistency and error bounds on coverage probability. Data-driven hyper-parameter choices are introduced only for practical feasibility and are not shown to enter the bound derivations as fitted inputs or self-definitions. No equations, self-citations, ansatzes, or renamings are quoted that would reduce any claimed prediction or bound to an input by construction. The central theoretical claims therefore retain independent content and do not match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only abstract available; ledger populated from stated elements.

free parameters (1)
  • hyper-parameters
    Data-driven choices proposed to guarantee practical feasibility; values not specified.
axioms (1)
  • domain assumption Optimal transport couplings exist and can be used to minimize coverage deviations from target levels
    Invoked when extending quantile-based intervals via optimal couplings.

pith-pipeline@v0.9.1-grok · 5682 in / 1088 out tokens · 21186 ms · 2026-06-26T11:44:10.187764+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

223 extracted references · 15 canonical work pages

  1. [1]

    2021 , doi =

    Ambrosio, Luigi and Bru\'e, Elia and Semola, Daniele , title =. 2021 , doi =

    2021

  2. [2]

    2001 , publisher =

    Combinatorial Methods in Density Estimation , author =. 2001 , publisher =

    2001

  3. [3]

    2013 , isbn =

    Stochastic Geometry and Its Applications , author =. 2013 , isbn =

    2013

  4. [4]

    , title =

    Bogachev, Vladimir I. , title =. 1998 , isbn =

    1998

  5. [5]

    Topics in Optimal Transportation , publisher =

    Villani, C. Topics in Optimal Transportation , publisher =. 2003 , isbn =

    2003

  6. [6]

    2021 , doi =

    Figalli, Alessio and Glaudo, Federico , title =. 2021 , doi =

    2021

  7. [7]

    Panaretos and Yoav Zemel

    Panaretos, Victor M. and Zemel, Yoav , title =. 2020 , isbn =. doi:10.1007/978-3-030-38438-8 , pages =

    work page doi:10.1007/978-3-030-38438-8 2020

  8. [8]

    , title =

    Rockafellar, Ralph T. , title =. 1970 , isbn =

    1970

  9. [9]

    1781 , note =

    Monge, Gaspard , title =. 1781 , note =

  10. [10]

    Wand, M. P. and Jones, M. C. , title =. 1995 , isbn =

    1995

  11. [11]

    , title =

    Li, Qi and Racine, Jeffrey S. , title =. 2007 , isbn =

    2007

  12. [12]

    2016 , isbn =

    Pratesi, Monica , title =. 2016 , isbn =

    2016

  13. [13]

    Optimal Transport: Old and New , series =

    Villani, C. Optimal Transport: Old and New , series =. 2009 , isbn =

    2009

  14. [14]

    Computational Optimal Transport , year =

    Peyr. Computational Optimal Transport , year =. 1803.00567 , archivePrefix=

    arXiv

  15. [15]

    2015 , doi =

    Santambrogio, Filippo , title =. 2015 , doi =

    2015

  16. [16]

    , title =

    Efron, Bradley and Tibshirani, Robert J. , title =. 1994 , doi =

    1994

  17. [17]

    1992 , doi =

    Hall, Peter , title =. 1992 , doi =

    1992

  18. [18]

    and Hinkley, David V

    Davison, Anthony C. and Hinkley, David V. , title =. 1997 , doi =

    1997

  19. [19]

    2003 , doi =

    Shao, Jun , title =. 2003 , doi =

    2003

  20. [20]

    , title =

    van der Vaart, Aad W. , title =. 1998 , doi =

    1998

  21. [21]

    , title =

    Liese, Friedrich and Miescke, Klaus-J. , title =. 2008 , doi =

    2008

  22. [22]

    Gradient Flows , subtitle =

    Ambrosio, Luigi and Gigli, Nicola and Savar. Gradient Flows , subtitle =. 2008 , doi =

    2008

  23. [23]

    2017 , edition=

    Random Measures, Theory and Applications , author=. 2017 , edition=

    2017

  24. [24]

    2004 , isbn =

    Boyd, Stephen and Vandenberghe, Lieven , title =. 2004 , isbn =

    2004

  25. [25]

    2013 , isbn =

    de Philippis, Guido , title =. 2013 , isbn =. doi:10.1007/978-88-7642-458-8 , pages =

    work page doi:10.1007/978-88-7642-458-8 2013

  26. [26]

    Lehmann and J

    Lehmann, Erich L. and Romano, Joseph P. , title =. 2022 , isbn =. doi:10.1007/978-3-030-70578-7 , pages =

    work page doi:10.1007/978-3-030-70578-7 2022

  27. [27]

    https://doi.org/10.1007/978-3-031-85160-5

    Chewi, Sinho and Niles-Weed, Jonathan and Rigollet, Philippe , title =. 2025 , isbn =. doi:10.1007/978-3-031-85160-5 , pages =

    work page doi:10.1007/978-3-031-85160-5 2025

  28. [28]

    , title =

    Dudley, Richard M. , title =. 2002 , isbn =. doi:10.1017/CBO9780511755347 , pages =

    work page doi:10.1017/cbo9780511755347 2002

  29. [29]

    2016 , isbn =

    Galichon, Alfred , title =. 2016 , isbn =

    2016

  30. [30]

    2023 , isbn =

    Nickl, Richard , title =. 2023 , isbn =

    2023

  31. [31]

    2005 , isbn =

    Kaipio, Jari and Somersalo, Erkki , title =. 2005 , isbn =

    2005

  32. [32]

    Concentration Inequalities: A Nonasymptotic Theory of Independence , edition =

    Boucheron, St. Concentration Inequalities: A Nonasymptotic Theory of Independence , edition =. 2013 , isbn =

    2013

  33. [33]

    , title =

    Wainwright, Martin J. , title =. 2019 , doi =

    2019

  34. [34]

    Rachev, Svetlozar T. and R. Mass Transportation Problem Volume I: Theory , series =. 1998 , isbn =

    1998

  35. [35]

    Rachev, Svetlozar T. and R. Mass Transportation Problem Volume II: Applications , series =. 1998 , isbn =

    1998

  36. [36]

    and Wellner, Jon A

    van der Vaart, Aad W. and Wellner, Jon A. , title =. 2023 , isbn =

    2023

  37. [37]

    2019 , doi =

    Bobkov, Sergey and Ledoux, Michel , title =. 2019 , doi =

    2019

  38. [38]

    and Wellner, Jon A

    Shorack, Galen R. and Wellner, Jon A. , title =. 2009 , isbn =

    2009

  39. [39]

    Sampling and Estimation from Finite Populations , series =

    Till. Sampling and Estimation from Finite Populations , series =. 2020 , doi =

    2020

  40. [40]

    Guidelines on small area estimation for city statistics and other functional geographies , year=

    Eurostat , publisher=. Guidelines on small area estimation for city statistics and other functional geographies , year=

  41. [41]

    doi:http://dx.doi.org/10.22617/TIM200160-2 , isbn=

    Introduction to Small Area Estimation Techniques: A Practical Guide for National Statistics Offices , year=. doi:http://dx.doi.org/10.22617/TIM200160-2 , isbn=

    work page doi:10.22617/tim200160-2

  42. [42]

    and Stern, Hal S

    Gelman, Andrew and Carlin, John B. and Stern, Hal S. and Rubin, Donald B. , title =. 2004 , doi =

    2004

  43. [43]

    , title =

    Lohr, Sharon L. , title =. 2019 , doi =

    2019

  44. [44]

    , title =

    Longford, Nicholas T. , title =. 2005 , isbn =

    2005

  45. [45]

    , title =

    Hochberg, Yosef and Tamhane, Ajit C. , title =. 1987 , isbn =

    1987

  46. [46]

    Rao, Jon N. K. and Molina, Isabel , title =. 2015 , isbn =

    2015

  47. [47]

    A Course on Small Area Estimation and Mixed Models , series =

    Morales, Domingo and Esteban, Mar. A Course on Small Area Estimation and Mixed Models , series =. 2022 , doi =

    2022

  48. [48]

    Model Assisted Survey Sampling , series =

    S. Model Assisted Survey Sampling , series =. 1992 , isbn =

    1992

  49. [49]

    Lehtonen, Risto and Veijanen, Ari , title =

  50. [50]

    , title =

    Cochran, William G. , title =. 1977 , isbn =

    1977

  51. [51]

    , title =

    Fuller, Wayne A. , title =. 2009 , isbn =

    2009

  52. [52]

    1968 , isbn =

    Raj, Des , title =. 1968 , isbn =

    1968

  53. [53]

    1974 , isbn =

    Mincer, Jacob , title =. 1974 , isbn =

    1974

  54. [54]

    2016 , url =

    van Handel, Ramon , title =. 2016 , url =

    2016

  55. [55]

    2001 , isbn =

    Ledoux, Michel , title =. 2001 , isbn =

    2001

  56. [56]

    High-Dimensional Probability: An Introduction with Applications in Data Science, volume 47 of Cambridge Series in Statistical and Probabilistic Mathematics

    Vershynin, Roman , title =. 2018 , isbn =. doi:10.1017/9781108231596 , url =

    work page doi:10.1017/9781108231596 2018

  57. [57]

    2011 , doi =

    Koltchinskii, Vladimir , title =. 2011 , doi =

    2011

  58. [58]

    and Wets, Roger J.-B

    Rockafellar, Ralph T. and Wets, Roger J.-B. , title =. 2009 , isbn =

    2009

  59. [59]

    , title =

    van de Geer, Sara A. , title =. 2009 , isbn =

    2009

  60. [60]

    Tsybakov.Introduction to Nonparametric Estimation

    Tsybakov, Alexandre B. , title =. 2009 , isbn =. doi:10.1007/b13794 , pages =

    work page doi:10.1007/b13794 2009

  61. [61]

    and Arsenin, Vasiliy Y

    Tikhonov, Andrey N. and Arsenin, Vasiliy Y. , title =. 1977 , isbn =

    1977

  62. [62]

    and Hanke, Martin and Neubauer, Andreas , title =

    Engl, Heinz W. and Hanke, Martin and Neubauer, Andreas , title =. 1996 , isbn =

    1996

  63. [63]

    1990 , isbn =

    Wahba, Grace , title =. 1990 , isbn =

    1990

  64. [64]

    and Silverman, Bernard W

    Green, Peter J. and Silverman, Bernard W. , title =. 1994 , isbn =

    1994

  65. [65]

    Statistics for High-Dimensional Data: Methods, Theory and Applications , edition =

    B. Statistics for High-Dimensional Data: Methods, Theory and Applications , edition =. 2011 , isbn =

    2011

  66. [66]

    , title =

    Dudley, Richard M. , title =. 1999 , doi =

    1999

  67. [67]

    1990 , isbn =

    Pollard, David , title =. 1990 , isbn =

    1990

  68. [68]

    Sampling Algorithms , series =

    Till. Sampling Algorithms , series =. 2006 , isbn =

    2006

  69. [69]

    and Casella, George and McCulloch, Charles E

    Searle, Shayle R. and Casella, George and McCulloch, Charles E. , title =. 1992 , isbn =

    1992

  70. [70]

    Brezis.Functional Analysis, Sobolev Spaces and Partial Differential Equations

    Brezis, Haim , title =. 2010 , isbn =. doi:10.1007/978-0-387-70914-7 , pages =

    work page doi:10.1007/978-0-387-70914-7 2010

  71. [71]

    and LEDOUX, M

    Bakry, Dominique and Gentil, Ivan and Ledoux, Michel , title =. 2014 , isbn =. doi:10.1007/978-3-319-00227-9 , pages =

    work page doi:10.1007/978-3-319-00227-9 2014

  72. [72]

    Le Cam and G.L

    Le Cam, Lucien and Yang, Grace L. , title =. 2000 , isbn =. doi:10.1007/978-1-4612-1166-2 , pages =

    work page doi:10.1007/978-1-4612-1166-2 2000

  73. [73]

    2021 , publisher =

    Linear and Generalized Linear Mixed Models and Their Applications , author =. 2021 , publisher =. doi:10.1007/978-1-0716-1282-8 , pages =

    work page doi:10.1007/978-1-0716-1282-8 2021

  74. [74]

    , title =

    Pavliotis, Grigorios A. , title =. 2014 , edition =

    2014

  75. [75]

    , title =

    Hsu, Jason C. , title =. 1996 , publisher =

    1996

  76. [76]

    Electronic Journal of Statistics , volume =

    Ponnoprat, Donlapark and Okano, Ryo and Imaizumi, Masaaki , title =. Electronic Journal of Statistics , volume =. 2024 , doi =

    2024

  77. [77]

    Proceedings of the 36th Conference on Learning Theory (COLT) , volume =

    Liu, Shuyu and Bunea, Florentina and Niles‑Weed, Jonathan , title =. Proceedings of the 36th Conference on Learning Theory (COLT) , volume =. 2023 , url =

    2023

  78. [78]

    , title =

    Kantorovitch, Leonid V. , title =. Doklady Akademii Nauk SSSR , volume =

  79. [79]

    The Annals of Statistics , volume =

    Efron, Bradley , title =. The Annals of Statistics , volume =. 1979 , doi =

    1979

  80. [80]

    Econometrica , volume =

    Foster, James and Greer, Joel and Thorbecke, Erik , title =. Econometrica , volume =

Showing first 80 references.