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Lift expectations of random sets
[Augmenter les attentes concernant les ensembles aléatoires]

Author

Listed:
  • Marc-Arthur Diaye

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Gleb Koshevoy
  • Ilya Molchanov

Abstract

It is known that the distribution of an integrable random vector ξ in Rd is uniquely determined by a (d + 1)-dimensional convex body called the lift zonoid of ξ. This concept is generalised to define the lift expectation of random convex bodies. However, the unique identification property of distributions is lost; it is shown that the lift expectation uniquely identifies only one-dimensional distributions of the support function, and so different random convex bodies may share the same lift expectation. The extent of this nonuniqueness is analysed and it is related to the identification of random convex functions using only their one- dimensional marginals. Applications to construction of depth-trimmed regions and partial ordering of random convex bodies are also mentioned.

Suggested Citation

  • Marc-Arthur Diaye & Gleb Koshevoy & Ilya Molchanov, 2019. "Lift expectations of random sets [Augmenter les attentes concernant les ensembles aléatoires]," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-03897964, HAL.
    • Handle: RePEc:hal:cesptp:hal-03897964
    DOI: 10.1016/j.spl.2018.08.015
    Note: View the original document on HAL open archive server: https://cnrs.hal.science/hal-03897964v1
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    References listed on IDEAS

    as
    1. Molchanov,Ilya & Molinari,Francesca, 2018. "Random Sets in Econometrics," Cambridge Books, Cambridge University Press, number 9781107121201, October.
    2. Puccetti, Giovanni & Scarsini, Marco, 2010. "Multivariate comonotonicity," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 291-304, January.
    3. Marco Dall’Aglio & Marco Scarsini, 2000. "Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex," ICER Working Papers - Applied Mathematics Series 27-2003, ICER - International Centre for Economic Research, revised Jul 2003.
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