Shape of a Distribution Through the L2-Wasserstein Distance

  1. Cuesta-Albertos, Juan A. 1
  2. Bea, Carlos Matrán 1
  3. Rodríguez, Jesús M. Rodríguez 1
  1. 1 Universidad de Valladolid
    info

    Universidad de Valladolid

    Valladolid, España

    ROR https://ror.org/01fvbaw18

Buch:
Distributions With Given Marginals and Statistical Modelling

ISBN: 9789048161362,  9789401700610

Datum der Publikation: 2002

Seiten: 51-61

Art: Buch-Kapitel

DOI: 10.1007/978-94-017-0061-0_7 GOOGLE SCHOLAR lock_openOpen Access editor

Zusammenfassung

Abstract Let Q be a probability measure on ℝd and let ℑ be a family of probability measures on ℝd which will be considered as a pattern. For suitable patterns we consider the closest law to Q in ℑ, through the L2-Wasserstein distance, as a descriptive measure associated to Q. The distance between Q and ℑ is a natural measure of the fit of Q to the pattern.We analyze this approach via the consideration of different patterns. Some of them generalize usual location and dispersion measures. Special attention will be paid to patterns based on uniform distributions on suitable families of sets, like the intervals in the unidimensional case (which allows us to analyze the flatness of the one-dimensional distributions) or the ellipsoids for the multivariate distributions.

Bibliographische Referenzen

  • Aurenhammer, F., F. Hoffmann, and B. Aronov (1998), ‘Minkowski-type theorems and leastsquares clustering’. Algorithmica 20, 61–76.
  • Bickel, P. and D. Freedman (1981), ‘Some asymptotic theory for the bootstrap’. Ann. Statist. 9, 1196–1217.
  • Cuesta-Albertos, J. A. and C. Matrán Bea (1989), ‘Notes on the Wasserstein metric in Hilbert spaces’. Ann. Probab. 17, 1264–1276.
  • Cuesta-Albertos, J. A., C. Matrán Bea, and A. Tuero Díaz (1996), ‘On lower bounds for the L2-Wasserstein metric in a Hilbert space’. J. Theor. Probab. 9, 263–283.
  • Cuesta-Albertos, J. A., C. Matrán Bea, and A. Tuero Díaz (1997), ‘Optimal transportation plans and convergence in distribution’. J. Multivariate Anal. 60, 72–83.
  • Cuesta-Albertos, J. A., L. Rüschendorf, and A. Tuero Díaz (1993), ‘Optimal coupling of multivariate distributions and stochastic processes’. J. Multivariate Anal. 46, 355–361.
  • Rachev, S. T. and L. Rüschendorf (1998), Mass transportation problems. Vol. I and II. SpringerVerlag, New York.
  • Tarpey, T. and B. Flury (1996), ‘Self-Consistency: A Fundamental Concept in Statistics’. Statist. Sci. 11, 229–243.