IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v87y2003i1p2-23.html
   My bibliography  Save this article

Multivariate quadratic forms of random vectors

Author

Listed:
  • Blacher, René

Abstract

We obtain the distribution of the sum of n random vectors and the distribution of their quadratic forms: their densities are expanded in series of Hermite and Laguerre polynomials. We do not suppose that these vectors are independent. In particular, we apply these results to multivariate quadratic forms of Gaussian vectors. We obtain also their densities expanded in Mac Laurin series or in the form of an integral. By this last result, we introduce a new method of computation which can be much simpler than the previously known techniques. In particular, we introduce a new method in the very classical univariate case. We remark that we do not assume the independence of normal variables.

Suggested Citation

  • Blacher, René, 2003. "Multivariate quadratic forms of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 2-23, October.
    • Handle: RePEc:eee:jmvana:v:87:y:2003:i:1:p:2-23
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(02)00013-1
    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. Mathai, A. M. & Moschopoulos, P. G., 1991. "On a multivariate gamma," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 135-153, October.
    2. Coelho, Carlos A., 1998. "The Generalized Integer Gamma Distribution--A Basis for Distributions in Multivariate Statistics," Journal of Multivariate Analysis, Elsevier, vol. 64(1), pages 86-102, January.
    3. Royen, T., 1991. "Expansions for the multivariate chi-square distribution," Journal of Multivariate Analysis, Elsevier, vol. 38(2), pages 213-232, August.
    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. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Post-Print hal-00800997, HAL.
    2. Matthieu Garcin & Dominique Guegan, 2013. "Probability density of the wavelet coefficients of a noisy chaos," Documents de travail du Centre d'Economie de la Sorbonne 13015, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    3. Song, Iickho & Lee, Seungwon, 2015. "Explicit formulae for product moments of multivariate Gaussian random variables," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 27-34.
    4. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    5. Regoli, Giuliana, 2009. "A class of bivariate exponential distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1261-1269, 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. 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.
    2. Buchmann, Boris & Kaehler, Benjamin & Maller, Ross & Szimayer, Alexander, 2017. "Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2208-2242.
    3. 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.
    4. Chiragiev, Arthur & Landsman, Zinoviy, 2009. "Multivariate flexible Pareto model: Dependency structure, properties and characterizations," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1733-1743, August.
    5. Alai, Daniel H. & Landsman, Zinoviy & Sherris, Michael, 2015. "A multivariate Tweedie lifetime model: Censoring and truncation," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 203-213.
    6. Coelho, Carlos A. & Marques, Filipe J., 2010. "Near-exact distributions for the independence and sphericity likelihood ratio test statistics," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 583-593, March.
    7. T. Royen, 1994. "On some multivariate gamma-distributions connected with spanning trees," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(2), pages 361-371, June.
    8. Arnold, Barry C. & Coelho, Carlos A. & Marques, Filipe J., 2013. "The distribution of the product of powers of independent uniform random variables — A simple but useful tool to address and better understand the structure of some distributions," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 19-36.
    9. A. Mathal & P. Moschopoulos, 1992. "A form of multivariate gamma distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(1), pages 97-106, March.
    10. Serim Hong & Carlos A. Coelho & Junyong Park, 2022. "An Exact and Near-Exact Distribution Approach to the Behrens–Fisher Problem," Mathematics, MDPI, vol. 10(16), pages 1-17, August.
    11. Alai, Daniel H. & Landsman, Zinoviy & Sherris, Michael, 2013. "Lifetime dependence modelling using a truncated multivariate gamma distribution," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 542-549.
    12. Yoshihide Kakizawa, 2009. "Multiple comparisons of several homoscedastic multivariate populations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(1), pages 1-26, March.
    13. Marques, Filipe J. & Loingeville, Florence, 2016. "Improved near-exact distributions for the product of independent Generalized Gamma random variables," Computational Statistics & Data Analysis, Elsevier, vol. 102(C), pages 55-66.
    14. Filipe Marques & Carlos Coelho, 2013. "Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions," Computational Statistics, Springer, vol. 28(5), pages 2091-2115, October.
    15. Dharmawansa, Prathapasinghe & McKay, Matthew R., 2009. "Diagonal distribution of a complex non-central Wishart matrix: A new trivariate non-central chi-squared density," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 561-580, April.
    16. Hagedorn, M. & Smith, P.J. & Bones, P.J. & Millane, R.P. & Pairman, D., 2006. "A trivariate chi-squared distribution derived from the complex Wishart distribution," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 655-674, March.
    17. T. Royen, 2007. "Integral Representations and Approximations for Multivariate Gamma Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 499-513, September.
    18. Zhou, Ming & Dhaene, Jan & Yao, Jing, 2018. "An approximation method for risk aggregations and capital allocation rules based on additive risk factor models," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 92-100.
    19. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
    20. Filipe J. Marques & Carlos A. Coelho & Paulo C. Rodrigues, 2017. "Testing the equality of several linear regression models," Computational Statistics, Springer, vol. 32(4), pages 1453-1480, December.

    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:87:y:2003:i:1:p:2-23. 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.