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Exact nonnull distribution of Wilks' statistic: The ratio and product of independent components

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  • Bekker, A.
  • Roux, J.J.J.
  • Arashi, M.

Abstract

The study of the noncentral matrix variate beta type distributions has been sidelined because the final expressions for the densities depend on an integral that has not been resolved in an explicit way. We derive an exact expression for the nonnull distribution of Wilks' statistic and precise expressions for the densities of the ratio and product of two independent components of matrix variates where one matrix variate has the noncentral matrix variate beta type I distribution and the other has the matrix variate beta type I distribution. We provide the expressions for the densities of the determinant of the ratio and the product of these two components. These distributions play a fundamental role in various areas of statistics, for example in the criteria proposed by Wilks.

Suggested Citation

  • Bekker, A. & Roux, J.J.J. & Arashi, M., 2011. "Exact nonnull distribution of Wilks' statistic: The ratio and product of independent components," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 619-628, March.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:3:p:619-628
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    References listed on IDEAS

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    1. T. Pham-Gia & N. Turkkan, 2002. "The product and quotient of general beta distributions," Statistical Papers, Springer, vol. 43(4), pages 537-550, October.
    2. Pham-Gia, T., 2008. "Exact distribution of the generalized Wilks's statistic and applications," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1698-1716, September.
    3. Y. Asoh & Masashi Okamoto, 1969. "A note on the non-null distribution of the Wilks statistic in MANOVA," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 21(1), pages 67-71, December.
    4. A. Gupta, 1975. "On a stochastic inequality for the wilks statistic," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 27(1), pages 341-348, December.
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    Cited by:

    1. Bekker, A. & Roux, J.J.J. & Ehlers, R. & Arashi, M., 2012. "Distribution of the product of determinants of noncentral bimatrix beta variates," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 73-87.

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