IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v151y2016icp90-109.html
   My bibliography  Save this article

Limit laws of the empirical Wasserstein distance: Gaussian distributions

Author

Listed:
  • Rippl, Thomas
  • Munk, Axel
  • Sturm, Anja

Abstract

We derive central limit theorems for the Wasserstein distance between the empirical distributions of Gaussian samples. The cases are distinguished whether the underlying laws are the same or different. Results are based on the (quadratic) Fréchet differentiability of the Wasserstein distance in the gaussian case. Extensions to elliptically symmetric distributions are discussed as well as several applications such as bootstrap and statistical testing.

Suggested Citation

  • Rippl, Thomas & Munk, Axel & Sturm, Anja, 2016. "Limit laws of the empirical Wasserstein distance: Gaussian distributions," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 90-109.
  • Handle: RePEc:eee:jmvana:v:151:y:2016:i:c:p:90-109
    DOI: 10.1016/j.jmva.2016.06.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X16300446
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.jmva.2016.06.005?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Dowson, D. C. & Landau, B. V., 1982. "The Fréchet distance between multivariate normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 12(3), pages 450-455, September.
    2. Rüschendorf, L. & Rachev, S. T., 1990. "A characterization of random variables with minimum L2-distance," Journal of Multivariate Analysis, Elsevier, vol. 32(1), pages 48-54, January.
    3. Major, Péter, 1978. "On the invariance principle for sums of independent identically distributed random variables," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 487-517, December.
    4. Freitag, Gudrun & Munk, Axel, 2005. "On Hadamard differentiability in k-sample semiparametric models--with applications to the assessment of structural relationships," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 123-158, May.
    5. Agulló-Antolín, Marina & Cuesta-Albertos, J.A. & Lescornel, Hélène & Loubes, Jean-Michel, 2015. "A parametric registration model for warped distributions with Wasserstein’s distance," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 117-130.
    6. Ruymgaart, Frits H. & Yang, Song, 1997. "Some Applications of Watson's Perturbation Approach to Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 48-60, January.
    7. Axel Munk & Claudia Czado, 1998. "Nonparametric validation of similar distributions and assessment of goodness of fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 223-241.
    8. Alvarez-Esteban, Pedro Cesar & del Barrio, Eustasio & Cuesta-Albertos, Juan Antonio & Matran, Carlos, 2008. "Trimmed Comparison of Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 697-704, June.
    9. Eustasio Barrio & Juan Cuesta-Albertos & Carlos Matrán & Sándor Csörgö & Carles Cuadras & Tertius Wet & Evarist Giné & Richard Lockhart & Axel Munk & Winfried Stute, 2000. "Contributions of empirical and quantile processes to the asymptotic theory of goodness-of-fit tests," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(1), pages 1-96, June.
    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. Valentina Masarotto & Victor M. Panaretos & Yoav Zemel, 2019. "Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 172-213, February.
    2. Mordant, Gilles & Segers, Johan, 2022. "Measuring dependence between random vectors via optimal transport," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    3. Valentin Hartmann & Dominic Schuhmacher, 2020. "Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 133-163, August.
    4. Viet Anh Nguyen & Daniel Kuhn & Peyman Mohajerin Esfahani, 2018. "Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator," Papers 1805.07194, arXiv.org.

    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. del Barrio, Eustasio & Gordaliza, Paula & Lescornel, Hélène & Loubes, Jean-Michel, 2019. "Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 341-362.
    2. Alvarez-Esteban, Pedro C. & del Barrio, Eustasio & Cuesta-Albertos, Juan A. & Matrán, Carlos, 2010. "Assessing when a sample is mostly normal," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 2914-2925, December.
    3. Freitag, Gudrun & Munk, Axel, 2005. "On Hadamard differentiability in k-sample semiparametric models--with applications to the assessment of structural relationships," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 123-158, May.
    4. E. Barrio & H. Inouzhe & C. Matrán, 2020. "On approximate validation of models: a Kolmogorov–Smirnov-based approach," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(4), pages 938-965, December.
    5. Puccetti, Giovanni & Rüschendorf, Ludger & Vanduffel, Steven, 2020. "On the computation of Wasserstein barycenters," Journal of Multivariate Analysis, Elsevier, vol. 176(C).
    6. Eustasio del Barrio & Hristo Inouzhe & Carlos Matrán, 2020. "Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 65-87, October.
    7. Xu, Ganggang & Zhu, Huirong & Lee, J. Jack, 2020. "Borrowing strength and borrowing index for Bayesian hierarchical models," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    8. L. Baringhaus & N. Henze, 2017. "Cramér–von Mises distance: probabilistic interpretation, confidence intervals, and neighbourhood-of-model validation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(2), pages 167-188, April.
    9. Knott, Martin & Smith, Cyril, 2006. "Choosing joint distributions so that the variance of the sum is small," Journal of Multivariate Analysis, Elsevier, vol. 97(8), pages 1757-1765, September.
    10. Jan Bergenthum & Ludger Rüschendorf, 2006. "Comparison of Option Prices in Semimartingale Models," Finance and Stochastics, Springer, vol. 10(2), pages 222-249, April.
    11. Menneteau, Ludovic, 2005. "Some laws of the iterated logarithm in Hilbertian autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 405-425, February.
    12. Elham Yousefi & Luc Pronzato & Markus Hainy & Werner G. Müller & Henry P. Wynn, 2023. "Discrimination between Gaussian process models: active learning and static constructions," Statistical Papers, Springer, vol. 64(4), pages 1275-1304, August.
    13. Munk, A. & Paige, R. & Pang, J. & Patrangenaru, V. & Ruymgaart, F., 2008. "The one- and multi-sample problem for functional data with application to projective shape analysis," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 815-833, May.
    14. Alfred Galichon & Damien Bosc, 2010. "Extreme dependence for multivariate data," Working Papers hal-03588294, HAL.
    15. P. C. Álvarez-Esteban & E. del Barrio & J. A. Cuesta-Albertos & C. Matrán, 2016. "A contamination model for the stochastic order," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 751-774, December.
    16. Mordant, Gilles & Segers, Johan, 2022. "Measuring dependence between random vectors via optimal transport," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    17. Rüschendorf, Ludger & Uckelmann, Ludger, 2002. "On the n-Coupling Problem," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 242-258, May.
    18. Yuan Hu & Abootaleb Shirvani & W. Brent Lindquist & Frank J. Fabozzi & Svetlozar T. Rachev, 2020. "Option Pricing Incorporating Factor Dynamics in Complete Markets," Papers 2011.08343, arXiv.org.
    19. Alfred Galichon & Damien Bosc, 2010. "Extreme dependence for multivariate data," SciencePo Working papers hal-03588294, HAL.
    20. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).

    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:eee:jmvana:v:151:y:2016:i:c:p:90-109. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.