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Distribution of the largest root of a matrix for Roy’s test in multivariate analysis of variance

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  • Chiani, Marco

Abstract

Let X,Y denote two independent real Gaussian p×m and p×n matrices with m,n≥p, each constituted by zero mean independent, identically distributed columns with common covariance. The Roy’s largest root criterion, used in multivariate analysis of variance (MANOVA), is based on the statistic of the largest eigenvalue, Θ1, of (A+B)−1B, where A=XXT and B=Y YT are independent central Wishart matrices. We derive a new expression and efficient recursive formulas for the exact distribution of Θ1. The expression can be easily calculated even for large parameters, eliminating the need of pre-calculated tables for the application of the Roy’s test.

Suggested Citation

  • Chiani, Marco, 2016. "Distribution of the largest root of a matrix for Roy’s test in multivariate analysis of variance," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 467-471.
    • Handle: RePEc:eee:jmvana:v:143:y:2016:i:c:p:467-471
    DOI: 10.1016/j.jmva.2015.10.007
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    References listed on IDEAS

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    1. Chiani, Marco, 2014. "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 69-81.
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    Cited by:

    1. Dharmawansa, Prathapasinghe & Nadler, Boaz & Shwartz, Ofer, 2019. "Roy’s largest root under rank-one perturbations: The complex valued case and applications," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    2. Takayama, Nobuki & Jiu, Lin & Kuriki, Satoshi & Zhang, Yi, 2020. "Computation of the expected Euler characteristic for the largest eigenvalue of a real non-central Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    3. Aurelia Rybak & Aleksandra Rybak & Jarosław Joostberens, 2023. "The Impact of Removing Coal from Poland’s Energy Mix on Selected Aspects of the Country’s Energy Security," Sustainability, MDPI, vol. 15(4), pages 1-13, February.
    4. Chételat, Didier & Narayanan, Rajendran & Wells, Martin T., 2018. "On the domain of attraction of a Tracy–Widom law with applications to testing multiple largest roots," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 132-142.

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