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Approximation to Probabilities Through Uniform Laws on Convex Sets

Author

Listed:
  • J. A. Cuesta-Albertos

    (Universidad de Cantabria)

  • C. Matrán

    (Universidad de Valladolid)

  • J. Rodríguez-Rodríguez

    (Universidad de Valladolid)

Abstract

Let P be a probability distribution on ∝ d and let $$C$$ be the family of the uniform probabilities defined on compact convex sets of ∝ d with interior non-empty. We prove that there exists a best approximation to P in $$C$$ , based on the L 2-Wasserstein distance. The approximation can be considered as the best representation of P by a convex set in the minimum squares setting, improving on other existent representations for the shape of a distribution. As a by-product we obtain properties related to the limit behavior and marginals of uniform distributions on convex sets which can be of independent interest.

Suggested Citation

  • J. A. Cuesta-Albertos & C. Matrán & J. Rodríguez-Rodríguez, 2003. "Approximation to Probabilities Through Uniform Laws on Convex Sets," Journal of Theoretical Probability, Springer, vol. 16(2), pages 363-376, April.
    • Handle: RePEc:spr:jotpro:v:16:y:2003:i:2:d:10.1023_a:1023518526754
    DOI: 10.1023/A:1023518526754
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    References listed on IDEAS

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    1. Cuesta-Albertos, J. A. & Matrán, C. & Tuero-Diaz, A., 1997. "Optimal Transportation Plans and Convergence in Distribution," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 72-83, January.
    2. 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.
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