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Hessian orders and multinormal distributions

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  • Arlotto, Alessandro
  • Scarsini, Marco

Abstract

Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex cone of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are equal and the difference of their covariance matrices belongs to the dual of . Then we show that the same conditions are also sufficient for multinormal random vectors. We study several particular cases of this general result.

Suggested Citation

  • Arlotto, Alessandro & Scarsini, Marco, 2009. "Hessian orders and multinormal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2324-2330, November.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:10:p:2324-2330
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    1. Moshe Shaked & J. Shanthikumar, 1990. "Parametric stochastic convexity and concavity of stochastic processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(3), pages 509-531, September.
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    9. Massimo Marinacci & Luigi Montrucchio, 2003. "Ultramodular functions," ICER Working Papers - Applied Mathematics Series 13-2003, ICER - International Centre for Economic Research.
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    Cited by:

    1. Chuancun Yin, 2019. "Stochastic Orderings of Multivariate Elliptical Distributions," Papers 1910.07158, arXiv.org, revised Nov 2019.
    2. Pan, Xiaoqing & Qiu, Guoxin & Hu, Taizhong, 2016. "Stochastic orderings for elliptical random vectors," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 83-88.
    3. Andrea Galeotti & Christian Ghiglinoy & Sanjeev Goyal, 2016. "Financial Linkages, Portfolio Choice and Systemic Risk," Cambridge Working Papers in Economics 1612, Faculty of Economics, University of Cambridge.
    4. Müller, Alfred & Scarsini, Marco, 2012. "Fear of loss, inframodularity, and transfers," Journal of Economic Theory, Elsevier, vol. 147(4), pages 1490-1500.
    5. Margaret Meyer & Bruno Strulovici, 2013. "The Supermodular Stochastic Ordering," Discussion Papers 1563, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    6. Amiri, Mehdi & Izadkhah, Salman & Jamalizadeh, Ahad, 2020. "Linear orderings of the scale mixtures of the multivariate skew-normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    7. Margaret Meyer & Bruno Strulovici, 2013. "Beyond Correlation: Measuring Interdependence Through Complementarities," Economics Series Working Papers 655, University of Oxford, Department of Economics.
    8. Mehdi Amiri & Narayanaswamy Balakrishnan & Abbas Eftekharian, 2022. "Hessian orderings of multivariate normal variance-mean mixture distributions and their applications in evaluating dependent multivariate risk portfolios," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 31(3), pages 679-707, September.
    9. Jonathan Ansari & Ludger Rüschendorf, 2018. "Ordering Results for Risk Bounds and Cost-efficient Payoffs in Partially Specified Risk Factor Models," Methodology and Computing in Applied Probability, Springer, vol. 20(3), pages 817-838, September.

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