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Goodman  has pointed out the applications of the distributional results of the complex multivariate normal statistical analysis. Khatri , has suggested the maximum latent root statistic for testing the reality of a covariance matrix. The joint distribution of the latent roots under certain null hypotheses can be written as, , ,
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 . 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  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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