Geodesic
On the Noncentral Distributions of the Second Largest Roots of Three Matrices in Multivariate Analysis
Canadian mathematical bulletin, Tome 13 (1970) no. 3, pp. 299-304

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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. 
Al-Ani, Sabri. On the Noncentral Distributions of the Second Largest Roots of Three Matrices in Multivariate Analysis. Canadian mathematical bulletin, Tome 13 (1970) no. 3, pp. 299-304. doi: 10.4153/CMB-1970-058-1
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     author = {Al-Ani, Sabri},
     title = {On the {Noncentral} {Distributions} of the {Second} {Largest} {Roots} of {Three} {Matrices} in {Multivariate} {Analysis}},
     journal = {Canadian mathematical bulletin},
     pages = {299--304},
     year = {1970},
     volume = {13},
     number = {3},
     doi = {10.4153/CMB-1970-058-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-058-1/}
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