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Lift expectations of random sets

Author

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  • Diaye, Marc-Arthur
  • Koshevoy, Gleb A.
  • Molchanov, Ilya

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

  • Diaye, Marc-Arthur & Koshevoy, Gleb A. & Molchanov, Ilya, 2019. "Lift expectations of random sets," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 110-117.
    • Handle: RePEc:eee:stapro:v:145:y:2019:i:c:p:110-117
    DOI: 10.1016/j.spl.2018.08.015
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    References listed on IDEAS

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    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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