IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v91y2004i2p143-160.html
   My bibliography  Save this article

Multivariate Lukacs theorem

Author

Listed:
  • Bobecka, Konstancja
  • Wesolowski, Jacek

Abstract

According to the celebrated Lukacs theorem, independence of quotient and sum of two independent positive random variables characterizes the gamma distribution. Rather unexpectedly, it appears that in the multivariate setting, the analogous independence condition does not characterize the multivariate gamma distribution in general, but is far more restrictive: it implies that the respective random vectors have independent or linearly dependent components. Our basic tool is a solution of a related functional equation of a quite general nature. As a side effect the form of the multivariate distribution with univariate Pareto conditionals is derived.

Suggested Citation

  • Bobecka, Konstancja & Wesolowski, Jacek, 2004. "Multivariate Lukacs theorem," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 143-160, November.
    • Handle: RePEc:eee:jmvana:v:91:y:2004:i:2:p:143-160
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(03)00158-1
    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. Griffiths, R. C., 1984. "Characterization of infinitely divisible multivariate gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 15(1), pages 13-20, August.
    2. Arnold, Barry C., 1987. "Bivariate distributions with pareto conditionals," Statistics & Probability Letters, Elsevier, vol. 5(4), pages 263-266, June.
    3. Shun-Hwa Li & Wen-Jang Huang & Mong-Na Huang, 1994. "Characterizations of the Poisson process as a renewal process via two conditional moments," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(2), pages 351-360, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ferrari, A. & Letac, G. & Tourneret, J.-Y., 2007. "Exponential families of mixed Poisson distributions," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1283-1292, July.

    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. Pérez-Abreu, Victor & Stelzer, Robert, 2014. "Infinitely divisible multivariate and matrix Gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 155-175.
    2. Chou, Chao-Wei & Huang, Wen-Jang, 2004. "On characterizations of the gamma and generalized inverse Gaussian distributions," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 381-388, October.
    3. Michel Denuit & Yang Lu, 2021. "Wishart‐gamma random effects models with applications to nonlife insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 88(2), pages 443-481, June.
    4. Wen-Jang Huang & Nan-Cheng Su, 2013. "Identification of power distribution mixtures through regression of exponentials," Statistical Papers, Springer, vol. 54(1), pages 227-241, February.
    5. Ramesh Gupta, 2011. "Bivariate odds ratio and association measures," Statistical Papers, Springer, vol. 52(1), pages 125-138, February.
    6. Rajib Dey & M. Ataharul Islam, 2017. "A conditional count model for repeated count data and its application to GEE approach," Statistical Papers, Springer, vol. 58(2), pages 485-504, June.
    7. Das, Sourish & Dey, Dipak K., 2010. "On Bayesian inference for generalized multivariate gamma distribution," Statistics & Probability Letters, Elsevier, vol. 80(19-20), pages 1492-1499, October.
    8. Boris Buchmann & Benjamin Kaehler & Ross Maller & Alexander Szimayer, 2015. "Multivariate Subordination using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing," Papers 1502.03901, arXiv.org, revised Oct 2016.
    9. Gupta, Ramesh C., 2001. "Reliability Studies of Bivariate Distributions with Pareto Conditionals," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 214-225, February.
    10. José María Sarabia & Vanesa Jordá & Faustino Prieto & Montserrat Guillén, 2020. "Multivariate Classes of GB2 Distributions with Applications," Mathematics, MDPI, vol. 9(1), pages 1-21, December.
    11. Arnold, Barry C. & Sarabia, José María, 2022. "Conditional specification of statistical models: Classical models, new developments and challenges," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    12. Ramesh C. Gupta, 2006. "Reliability studies of bivariate distributions with Pearson type VII conditionals," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 239-251.
    13. Eisenbaum, Nathalie & Kaspi, Haya, 2009. "On permanental processes," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1401-1415, May.
    14. Denuit, Michel & Lu, Yang, 2020. "Wishart-Gamma mixtures for multiperil experience ratemaking, frequency-severity experience rating and micro-loss reserving," LIDAM Discussion Papers ISBA 2020016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Jesper Møller & Ege Rubak, 2010. "A Model for Positively Correlated Count Variables," International Statistical Review, International Statistical Institute, vol. 78(1), pages 65-80, April.
    16. S. M. Sunoj & N. Vipin, 2019. "Some properties of conditional partial moments in the context of stochastic modelling," Statistical Papers, Springer, vol. 60(6), pages 1971-1999, December.
    17. Sarabia, José María & Gómez-Déniz, Emilio & Prieto, Faustino & Jordá, Vanesa, 2016. "Risk aggregation in multivariate dependent Pareto distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 154-163.
    18. Navarro, Jorge & Sarabia, José María, 2013. "Reliability properties of bivariate conditional proportional hazard rate models," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 116-127.

    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:jmvana:v:91:y:2004:i:2:p:143-160. See general information about how to correct material in RePEc.

    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 bibliographic 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.

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

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.