IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v153y2019icp104-107.html
   My bibliography  Save this article

The Minkowski length of a spherical random vector

Author

Listed:
  • Shushi, Tomer

Abstract

The Rayleigh distribution represents the Euclidean length of a two-dimensional random vector with normally distributed components that are independent, while for the case of a three-dimensional random vector, its length distributed by the well-known Maxwell–Boltzmann distribution. In this letter, we generalize these results in two ways, into the world of elliptical distributions and for general Lp spaces. We present the distribution of the length of an n-dimensional random vector whose components are mutually dependent and symmetrically distributed in Lp spaces. The results show that such distribution has explicit form, which allows computing its moments. Similar to the Rayleigh distribution, the presented distribution can also be useful to model risks. Thus, we derive important risk measures for the investigated distribution.

Suggested Citation

  • Shushi, Tomer, 2019. "The Minkowski length of a spherical random vector," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 104-107.
    • Handle: RePEc:eee:stapro:v:153:y:2019:i:c:p:104-107
    DOI: 10.1016/j.spl.2019.06.003
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715219301555
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    2. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Tail conditional moments for elliptical and log-elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 179-188.
    3. McNeil, Alexander J. & Frey, Rudiger, 2000. "Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach," Journal of Empirical Finance, Elsevier, vol. 7(3-4), pages 271-300, November.
    4. Shushi, Tomer, 2018. "Stein’s lemma for truncated elliptical random vectors," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 297-303.
    5. Guy Kaplanski & Haim Levy, 2015. "Value-at-risk capital requirement regulation, risk taking and asset allocation: a mean-variance analysis," The European Journal of Finance, Taylor & Francis Journals, vol. 21(3), pages 215-241, February.
    6. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    7. Furman, Edward & Landsman, Zinoviy, 2006. "Tail Variance Premium with Applications for Elliptical Portfolio of Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 433-462, November.
    8. Tarpey, Thaddeus & Loperfido, Nicola, 2015. "Self-consistency and a generalized principal subspace theorem," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 27-37.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Chuancun Yin, 2019. "Stochastic ordering of Gini indexes for multivariate elliptical random variables," Papers 1908.01943, arXiv.org, revised Sep 2019.
    2. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Tail conditional moments for elliptical and log-elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 179-188.
    3. Nicole Bauerle & Tomer Shushi, 2019. "Risk Management with Tail Quasi-Linear Means," Papers 1902.06941, arXiv.org, revised Jan 2020.
    4. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2018. "A multivariate tail covariance measure for elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 27-35.
    5. Jaworski, Piotr & Pitera, Marcin, 2017. "A note on conditional covariance matrices for elliptical distributions," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 230-235.
    6. Piotr Jaworski & Marcin Pitera, 2017. "A note on conditional covariance matrices for elliptical distributions," Papers 1703.00918, arXiv.org.
    7. Yugu Xiao & Emiliano A. Valdez, 2015. "A Black-Litterman asset allocation model under Elliptical distributions," Quantitative Finance, Taylor & Francis Journals, vol. 15(3), pages 509-519, March.
    8. Kim, Joseph H.T. & Kim, So-Yeun, 2019. "Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 145-157.
    9. Nawaf Mohammed & Edward Furman & Jianxi Su, 2021. "Some results on the risk capital allocation rule induced by the Conditional Tail Expectation risk measure," Papers 2102.05003, arXiv.org, revised Feb 2021.
    10. Zaichao Du & Juan Carlos Escanciano, 2017. "Backtesting Expected Shortfall: Accounting for Tail Risk," Management Science, INFORMS, vol. 63(4), pages 940-958, April.
    11. Berens, Tobias & Weiß, Gregor N.F. & Wied, Dominik, 2015. "Testing for structural breaks in correlations: Does it improve Value-at-Risk forecasting?," Journal of Empirical Finance, Elsevier, vol. 32(C), pages 135-152.
    12. Alex Huang, 2013. "Value at risk estimation by quantile regression and kernel estimator," Review of Quantitative Finance and Accounting, Springer, vol. 41(2), pages 225-251, August.
    13. Falk, Michael, 1998. "A Note on the Comedian for Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 306-317, November.
    14. Xi, Yanhui & Peng, Hui & Qin, Yemei & Xie, Wenbiao & Chen, Xiaohong, 2015. "Bayesian analysis of heavy-tailed market microstructure model and its application in stock markets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 117(C), pages 141-153.
    15. Dimitrakopoulos, Dimitris N. & Kavussanos, Manolis G. & Spyrou, Spyros I., 2010. "Value at risk models for volatile emerging markets equity portfolios," The Quarterly Review of Economics and Finance, Elsevier, vol. 50(4), pages 515-526, November.
    16. Kume, Alfred & Hashorva, Enkelejd, 2012. "Calculation of Bayes premium for conditional elliptical risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 632-635.
    17. Furman, Edward & Landsman, Zinoviy, 2010. "Multivariate Tweedie distributions and some related capital-at-risk analyses," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 351-361, April.
    18. Jacob, P. & Suquet, Ch., 1997. "Regression and asymptotical location of a multivariate sample," Statistics & Probability Letters, Elsevier, vol. 35(2), pages 173-179, September.
    19. Marco Rocco, 2011. "Extreme value theory for finance: a survey," Questioni di Economia e Finanza (Occasional Papers) 99, Bank of Italy, Economic Research and International Relations Area.
    20. Luo, Weiwei & Brooks, Robert D. & Silvapulle, Param, 2011. "Effects of the open policy on the dependence between the Chinese 'A' stock market and other equity markets: An industry sector perspective," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 21(1), pages 49-74, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:153:y:2019:i:c:p:104-107. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Haili He). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.