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Expansions for the multivariate chi-square distribution

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  • Royen, T.

Abstract

Three classes of expansions for the distribution function of the [chi]k2(d, R)-distribution are given, where k denotes the dimension, d the degree of freedom, and R the "accompanying correlation matrix." The first class generalizes the orthogonal series with generalized Laguerre polynomials, originally given by Krishnamoorthy and Parthasarathy [12]. The second class contains always absolutely convergent representations of the distribution function by univariate chi-square distributions and the third class provides also the probabilities for any unbounded rectangular regions. In particular, simple formulas are given for the three-variate case including singular correlation matrices R, which simplify the computation of third order Bonferroni inequalities, e.g., for the tail probabilities of max{[chi]i21 3).

Suggested Citation

  • Royen, T., 1991. "Expansions for the multivariate chi-square distribution," Journal of Multivariate Analysis, Elsevier, vol. 38(2), pages 213-232, August.
    • Handle: RePEc:eee:jmvana:v:38:y:1991:i:2:p:213-232
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    Cited by:

    1. Nurulkamal Masseran, 2021. "Modeling the Characteristics of Unhealthy Air Pollution Events: A Copula Approach," IJERPH, MDPI, vol. 18(16), pages 1-18, August.
    2. 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.
    3. 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.
    4. 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.
    5. Blacher, René, 2003. "Multivariate quadratic forms of random vectors," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 2-23, October.
    6. 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.
    7. 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.

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