IDEAS home Printed from https://ideas.repec.org/p/eca/wpaper/2013-271399.html
   My bibliography  Save this paper

Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions

Author

Listed:
  • Eustasio Del Barrio
  • Juan Cuesta Albertos
  • Marc Hallin
  • Carlos Matran

Abstract

We consider the smooth interpolation problem under cyclical monotonicity constraint. More precisely, consider finite n-tuples X =fx1; xng and Y = fy1; yng of points in Rd, and assume the existence of a unique bijection T :X !Y such that f(x; T(x)): x 2 Xg is cyclically monotone: our goal is to define continuous, cyclically mono-tone maps T :Rd !Rd such that T(xi) = yi, i = 1; n, extending a classical result by Rockafellar on the sub differentials of convex functions. Our solutions T are Lipschitz, and we provide a sharp lower bound for the corresponding Lipschitz constants. The problem is motivated by, and the solution naturally applies to, the concept of empirical center-outwarddistribution function in Rd developed in Hallin (2018). Those empirical distribution functions indeed are de_ned at the observations only. Our interpolation provides a smooth extension, as well as a multivariate, outward-continuous, jump function version thereof (the latter naturally generalizes the traditional left-continuous univariate concept); both satisfy a Glivenko-Cantelli property as n !1.

Suggested Citation

  • Eustasio Del Barrio & Juan Cuesta Albertos & Marc Hallin & Carlos Matran, 2018. "Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions," Working Papers ECARES 2018-15, ULB -- Universite Libre de Bruxelles.
  • Handle: RePEc:eca:wpaper:2013/271399
    as

    Download full text from publisher

    File URL: https://dipot.ulb.ac.be/dspace/bitstream/2013/271399/3/2018-15-DELBARRIO_CUESTASALBERTO_HALLIN_MATRAN-smooth.pdf
    File Function: Œuvre complète ou partie de l'œuvre
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    2. Marc Hallin, 2017. "On Distribution and Quantile Functions, Ranks and Signs in R_d," Working Papers ECARES ECARES 2017-34, ULB -- Universite Libre de Bruxelles.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. de Valk, Cees Fouad & Segers, Johan, 2018. "Stability and tail limits of transport-based quantile contours," LIDAM Discussion Papers ISBA 2018031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Eustasio Del Barrio & Alberto Gonzalez-Sanz & Marc Hallin, 2019. "A Note on the Regularity of Center-Outward Distribution and Quantile Functions," Working Papers ECARES 2019-33, ULB -- Universite Libre de Bruxelles.
    3. Marc Hallin, 2021. "Measure Transportation and Statistical Decision Theory," Working Papers ECARES 2021-04, ULB -- Universite Libre de Bruxelles.
    4. Hongjian Shi & Marc Hallin & Mathias Drton & Fang Han, 2020. "Rate-Optimality of Consistent Distribution-Free Tests of Independence Based on Center-Outward Ranks and Signs," Working Papers ECARES 2020-23, ULB -- Universite Libre de Bruxelles.
    5. Marc Hallin, 2018. "From Mahalanobis to Bregman via Monge and Kantorovich," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 135-146, December.
    6. Beirlant, J. & Buitendag, S. & del Barrio, E. & Hallin, M. & Kamper, F., 2020. "Center-outward quantiles and the measurement of multivariate risk," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 79-100.
    7. Marc Hallin, 2018. "From Mahalanobis to Bregman via Monge and Kantorovich towards a “General Generalised Distance”," Working Papers ECARES 2018-12, ULB -- Universite Libre de Bruxelles.
    8. del Barrio, Eustasio & González-Sanz, Alberto & Hallin, Marc, 2020. "A note on the regularity of optimal-transport-based center-outward distribution and quantile functions," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    9. Hongjian Shi & Mathias Drton & Marc Hallin & Fang Han, 2023. "Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics," Working Papers ECARES 2023-03, ULB -- Universite Libre de Bruxelles.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. M. Hallin & D. La Vecchia & H. Liu, 2022. "Center-Outward R-Estimation for Semiparametric VARMA Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 117(538), pages 925-938, April.
    2. Hongjian Shi & Mathias Drton & Marc Hallin & Fang Han, 2023. "Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics," Working Papers ECARES 2023-03, ULB -- Universite Libre de Bruxelles.
    3. Hongjian Shi & Marc Hallin & Mathias Drton & Fang Han, 2020. "Rate-Optimality of Consistent Distribution-Free Tests of Independence Based on Center-Outward Ranks and Signs," Working Papers ECARES 2020-23, ULB -- Universite Libre de Bruxelles.
    4. Marc Hallin & Hongjian Shi & Mathias Drton & Fang Han, 2021. "Center-Outward Sign- and Rank-Based Quadrant, Spearman, and Kendall Tests for Multivariate Independence," Working Papers ECARES 2021-27, ULB -- Universite Libre de Bruxelles.
    5. Marc Hallin, 2018. "From Mahalanobis to Bregman via Monge and Kantorovich towards a “General Generalised Distance”," Working Papers ECARES 2018-12, ULB -- Universite Libre de Bruxelles.
    6. Hallin, Marc & La Vecchia, Davide, 2020. "A Simple R-estimation method for semiparametric duration models," Journal of Econometrics, Elsevier, vol. 218(2), pages 736-749.
    7. María Edo & Walter Sosa Escudero & Marcela Svarc, 2021. "A multidimensional approach to measuring the middle class," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 19(1), pages 139-162, March.
    8. Faugeras, Olivier P. & Rüschendorf, Ludger, 2021. "Functional, randomized and smoothed multivariate quantile regions," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    9. Marc Hallin & Gilles Mordant, 2021. "On the Finite-Sample Performance of Measure Transportation-Based Multivariate Rank Tests," Working Papers ECARES 2021-24, ULB -- Universite Libre de Bruxelles.
    10. Alfred Galichon & Bernard Salani'e, 2021. "Cupid's Invisible Hand: Social Surplus and Identification in Matching Models," Papers 2106.02371, arXiv.org, revised Jan 2023.
    11. de Valk, Cees Fouad & Segers, Johan, 2018. "Stability and tail limits of transport-based quantile contours," LIDAM Discussion Papers ISBA 2018031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    12. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    13. Marc Hallin & Daniel Hlubinka & Šárka Hudecová, 2023. "Efficient Fully Distribution-Free Center-Outward Rank Tests for Multiple-Output Regression and MANOVA," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(543), pages 1923-1939, July.
    14. Florian Gunsilius, 2018. "Point-identification in multivariate nonseparable triangular models," Papers 1806.09680, arXiv.org.
    15. Gunsilius, Florian F., 2023. "A condition for the identification of multivariate models with binary instruments," Journal of Econometrics, Elsevier, vol. 235(1), pages 220-238.
    16. Petra Laketa & Stanislav Nagy, 2022. "Halfspace depth for general measures: the ray basis theorem and its consequences," Statistical Papers, Springer, vol. 63(3), pages 849-883, June.
    17. Alberto González-Sanz & Marc Hallin & Bodhisattva Sen, 2023. "Monotone Measure-Preserving Maps in Hilbert Spaces: Existence, Uniqueness, and Stability," Working Papers ECARES 2023-10, ULB -- Universite Libre de Bruxelles.
    18. Beirlant, J. & Buitendag, S. & del Barrio, E. & Hallin, M. & Kamper, F., 2020. "Center-outward quantiles and the measurement of multivariate risk," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 79-100.
    19. Manuel Arellano & Stéphane Bonhomme, 2023. "Recovering Latent Variables by Matching," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(541), pages 693-706, January.
    20. Florian Gunsilius & Susanne M. Schennach, 2019. "Independent nonlinear component analysis," CeMMAP working papers CWP46/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eca:wpaper:2013/271399. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Benoit Pauwels (email available below). General contact details of provider: https://edirc.repec.org/data/arulbbe.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.