Estimators for multivariate allometric regression model
针对多元异速回归模型，提出基于加权平方和矩阵的第一主特征向量估计量类，推导均方误差上界并给出最优权重，在弱可识别性和高维大样本条件下讨论估计量的一致性。
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.