The Asymptotic Distribution of Characteristic Roots and Vectors in Multivariate Components of Variance

The asymptotic distribution of the characteristic roots and vectors of one Wishart matrix in the metric of another as the two degrees of freedom increase in fixed proportion is obtained. In the balanced one-way multivariate analysis of variance these two matrices are the sample effect and error covariance matrices, and the numbers of degrees of freedom are (approximately) proportional to the number of classes. The maximum likelihood estimate of the effect covariance matrix of a given rank depends on the characteristic roots and vectors.