In a regression model with multiple response variables and multiple explanatory variables, if the difference of the mean vectors of the response variables for different values of explanatory variables is always in the direction of the first principal eigenvector of the covariance matrix of the response variables, then it is called a multivariate allometric regression model. This paper studies the estimation of the first principal eigenvector in the multivariate allometric regression model. A class of estimators that includes conventional estimators is proposed based on weighted sum-of-squares matrices of regression sum-of-squares matrix and residual sum-of-squares matrix. We establish an upper bound of the mean squared error of the estimators contained in this class, and the weight value minimizing the upper bound is derived. Sufficient conditions for the consistency of the estimators are discussed in weak identifiability regimes under which the difference of the largest and second largest eigenvalues of the covariance matrix decays asymptotically and in ``large $p$, large $n$" regimes, where $p$ is the number of response variables and $n$ is the sample size. Several numerical results are also presented.

翻译：在包含多个响应变量和多个解释变量的回归模型中，若对于不同解释变量取值，响应变量均值向量的差异始终指向响应变量协方差矩阵的第一主特征向量方向，则称该模型为多元异速回归模型。本文研究该模型中第一主特征向量的估计问题。基于回归平方和矩阵与残差平方和矩阵的加权平方和矩阵，提出包含传统估计量的估计量族。建立该族估计量均方误差的上界，并推导出使该上界最小化的权重值。在弱可识别性条件下讨论估计量一致性的充分条件：该条件包括协方差矩阵最大与次大特征值之差渐近衰减的情形，以及"大规模$p$、大样本$n$"框架（其中$p$为响应变量数，$n$为样本量）。文中还呈现了若干数值结果。