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：在具有多个响应变量和多个解释变量的回归模型中，如果响应变量均值向量随解释变量取值不同而产生的差异始终位于响应变量协方差矩阵的第一主特征向量方向上，则该模型称为多元等距回归模型。本文研究多元等距回归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，提出了一类包含传统估计量的估计量。我们建立了该类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（即协方差矩阵最大特征值与第二大特征值之差渐近衰减）以及“大$p$，大$n$”条件下（其中$p$为响应变量个数，$n$为样本量），讨论了估计量一致性的充分条件。文中还给出了若干数值结果。