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Cone distribution functions and quantiles for multivariate random variables

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  • Hamel, Andreas H.
  • Kostner, Daniel

Abstract

Set-valued quantiles for multivariate distributions with respect to a general convex cone are introduced which are based on a family of (univariate) distribution functions rather than on the joint distribution function. It is shown that these quantiles enjoy basically all the properties of univariate quantile functions. Relationships to families of univariate quantile functions and to depth functions are discussed. Finally, a corresponding Value-at-Risk for multivariate random variables as well as a stochastic (dominance) order based on quantiles are introduced via the set-valued approach.

Suggested Citation

  • Hamel, Andreas H. & Kostner, Daniel, 2018. "Cone distribution functions and quantiles for multivariate random variables," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 97-113.
    • Handle: RePEc:eee:jmvana:v:167:y:2018:i:c:p:97-113
    DOI: 10.1016/j.jmva.2018.04.004
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Hamel, Andreas H. & Kostner, Daniel, 2022. "Computation of quantile sets for bivariate ordered data," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    2. Andreas H Hamel, 2018. "Monetary Measures of Risk," Papers 1812.04354, arXiv.org.
    3. Kuntal Som & V. Vetrivel, 2021. "On robustness for set-valued optimization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 905-925, April.
    4. Andreas H Hamel & Andreas Löhne, 2020. "Choosing sets: preface to the special issue on set optimization and applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 1-4, February.
    5. Elisa Mastrogiacomo & Matteo Rocca, 2021. "Set optimization of set-valued risk measures," Annals of Operations Research, Springer, vol. 296(1), pages 291-314, January.
    6. Tobias Fissler & Jana Hlavinov'a & Birgit Rudloff, 2019. "Elicitability and Identifiability of Systemic Risk Measures," Papers 1907.01306, arXiv.org, revised Oct 2019.
    7. Giovanni Paolo Crespi & Andreas H. Hamel & Matteo Rocca & Carola Schrage, 2021. "Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 361-381, February.
    8. Daniel Kostner, 2020. "Multi-criteria decision making via multivariate quantiles," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 73-88, February.
    9. Andreas H. Hamel & Frank Heyde, 2021. "Set-Valued T -Translative Functions and Their Applications in Finance," Mathematics, MDPI, vol. 9(18), pages 1-33, September.

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