Homogeneous matrix equations and multivariate linear models (Q1318199)

This paper presents a set of theorems on the restrictions imposed by the matrix equations \(DBE=0\), \(D_ i BE_ i=0\), \(i=1,2,\dots,s\) and \(D_ 1 B_ 1 E_ 1 + D_ 2 B_ 2 E_ 2=0\) in the multivariate parameter space of linear statistical models. In these homogeneous matrix equations the \(D\)'s and \(E\)'s are known matrices while the \(B\)'s are unknown. Let the columns of matrix \(X\) be independently normally distributed with an unknown dispersion matrix and \(E[x]=\sum^ m_{i=1} A_ iB_ iC_ i\), where the \(A\)'s and \(C\)'s are known and the \(B\)'s unknown. For this linear model the maximum likelihood estimators for the parameters in \(B\) are discussed. The conditions under which the above homogeneous restrictions are unique are also discussed and the connections with the growth model problems presented.