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On the Noncentral Distributions of the Second Largest Roots of Three Matrices in Multivariate Analysis

  • Sabri Al-Ani (a1)

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The central distribution of the second largest (smallest) root following the Fisher-Girshick-Hsu-Roy distribution under certain null-hypothesis has been derived in series form by Pillai and Al-Ani [6]. In this paper the noncentral distributions of the second largest roots in the MANOVA situation, the canonical correlation, and equality of two covariance matrices are obtained. Further, the distribution of the second largest root of the covariance matrix is obtained as a limiting case. The largest root and its noncentral distributions have been considered already by Pillai and Sugiyama [7] for the situations stated above. However, in the present paper, the joint densities of the largest and the second largest roots are derived in all the above cases from which the distributions of the largest roots can be obtained, although in more elaborate forms.

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Copyright

Footnotes

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(1)

This research was supported by the NSF Grant GP 7663, and this paper was written when the author was at Purdue University.

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References

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1. Constantine, A. G., Some non-central distribution problems in multivariate analysis, Ann. Math. Statist. 34 (1963), 1270-1285.
2. Hayakawa, T., On the distribution of the maximum latent root of a positive definite symmetric random matrix, Ann. Inst. Statist. Math. 19 (1967), 1-17.
3. James, A. T., The distribution of the latent roots of the covariance matrix, Ann. Math. Statist. 31 (1960), 151-158.
4. Khatri, C. G., Some distribution problems connected with the characteristic root of S1S2 -1 , Ann. Math. Statist. 38 (1967), 944-948.
5. Khatri, C. G., and Pillai, K. C. S., On the non-central distribution of two test criteria in multivariate analysis of variance, Ann. Math. Statist. 39, 1968.(to appear).
6. Pillai, K. C. S. and Al-Ani, S., On some distribution problems concerning characteristic roots and vectors in multivariate analysis. Mimeo. Series 123, Dept. of Statistics, Purdue University (1967).
7. Pillai, K. C. S. and Sugiyama, T., Non-central distributions of the largest latent roots of three matrices in multivariate analysis, Mimeo. Series 129, Dept. of Statistics, Purdue University, 1967.
8. Sugiyama, T., On the distribution of the largest latent root of the covariance matrix, Ann. Math. Statist. 38 (1967), 1148-1151.
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On the Noncentral Distributions of the Second Largest Roots of Three Matrices in Multivariate Analysis

  • Sabri Al-Ani (a1)

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