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Calculation of the Prokhorov distance by optimal quantization and maximum flow

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Abstract

Calculating exact values of the Prokhorov metric for the set of probability distributions on a metric space is a challenging problem. In this paper probability distributions are approximated by finite-support distributions through optimal or quasi-optimal quantization, in such a way that exact calculation of the Prokhorov distance between a distribution and a quantizer can be performed. The exact value of the Prokhorov distance between two quantizers is obtained by solving an optimization problem through the Simplex method. This last value is used to approximate the Prokhorov distance between the two initial distributions, and the accuracy of the approximation is measured. We illustrate the method on various univariate and bivariate probability distributions. Approximation of bivariate standard normal distributions by quasi-optimal quantizers is also considered.

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Authors and Affiliations

  1. Mathematical Institute of Toulouse (UPS) and ENSEEIHT, 2 rue Camichel, 31071, Toulouse Cédex 7, France

    Bernard Garel

  2. Département de Mathématiques et de Statistique, Université Laval, Sainte-Foy, Québec, G1K 7P4, Canada

    Jean-Claude Massé

Authors
  1. Bernard Garel
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  2. Jean-Claude Massé
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Correspondence to Bernard Garel.

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Garel, B., Massé, JC. Calculation of the Prokhorov distance by optimal quantization and maximum flow. AStA Adv Stat Anal 93, 73–88 (2009). https://doi.org/10.1007/s10182-008-0082-1

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  • DOI: https://doi.org/10.1007/s10182-008-0082-1

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