Asymptotic distribution for the proportional covariance model under multivariate normal distributions is derived. To this end, the parametrization of the common covariance matrix by its Cholesky root is adopted. The derivations are made in three steps. First, the asymptotic distribution of the maximum likelihood estimators of the proportionality coefficients and the Cholesky inverse root of the common covariance matrix is derived by finding the information matrix and its inverse. Next, the asymptotic distributions for the case of the Cholesky root of the common covariance matrix and finally for the case of the common covariance matrix itself are derived using the multivariate $\delta$-method. As an application of the asymptotic distribution derived here, a hypothesis for homogeneity of covariance matrices is considered.

翻译：多变量正常分布下比例共变模式的均衡共变分布。 为此, 将采用Cholesky root 的通用共变矩阵的平衡化法。 衍生出三个步骤。 首先, 通过查找信息矩阵及其反向, 得出相称系数和共同共变矩阵最大概率估计值的无平衡分布法。 其次, 共同共变矩阵的Choolesky根基和共同共变矩阵本身的无平衡分布法, 使用多变量 $\delta$- method 来得出。 作为此处得出的非均值分布法的应用, 考虑了共变矩阵同性假设 。