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Rates of convergence for partial mass problems
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  • Published: 09 December 2011

Rates of convergence for partial mass problems

  • Eustasio del Barrio1 &
  • Carlos Matrán1 

Probability Theory and Related Fields volume 155, pages 521–542 (2013)Cite this article

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Abstract

We consider a class of partial mass problems in which a fraction of the mass of a probability measure is allowed to be changed (trimmed) to maximize fit to a given pattern. This includes the problem of optimal partial transportation of mass, where a part of the mass need not be transported, and also trimming procedures which are often used in statistical data analysis to discard outliers in a sample (the data with lowest agreement to a certain pattern). This results in a modified, trimmed version of the original probability which is closer to the pattern. We focus on the case of the empirical measure and analyze to what extent its optimally trimmed version is closer to the true random generator in terms of rates of convergence. We deal with probabilities on \({\mathbb{R}^k}\) and measure agreement through probability metrics. Our choices include transportation cost metrics, associated to optimal partial transportation, and the Kolmogorov distance. We show that partial transportation (as opposed to classical, complete transportation) results in a sharp decrease of costs only in low dimension. In contrast, for the Kolmogorov metric this decrease is seen in any dimension.

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Authors and Affiliations

  1. IMUVA, Instituto de Matemáticas, Universidad de Valladolid, Prado de la Magdalena, S/N, 47005, Valladolid, Spain

    Eustasio del Barrio & Carlos Matrán

Authors
  1. Eustasio del Barrio
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  2. Carlos Matrán
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Corresponding author

Correspondence to Eustasio del Barrio.

Additional information

Research partially supported by the Spanish Ministerio de Educación y Ciencia and FEDER, Grant MTM2011-28657-C02-01.

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Cite this article

del Barrio, E., Matrán, C. Rates of convergence for partial mass problems. Probab. Theory Relat. Fields 155, 521–542 (2013). https://doi.org/10.1007/s00440-011-0406-z

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  • Received: 12 January 2011

  • Revised: 25 November 2011

  • Published: 09 December 2011

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-011-0406-z

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Keywords

  • Partial mass transportation problem
  • Random quantization
  • Optimal transportation plan
  • Similarity
  • Trimming
  • Trimmed probability
  • Kolmogorov distance
  • Wasserstein distance
  • Rate of convergence
  • Concentration of measure

Mathematics Subject Classification (2000)

  • Primary 60B10
  • Secondary 05C70
  • 60C05
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