The general linear model of the generalized singular value decomposition (Q1072313)

The following linear model is considered: \[ y=X\beta +w,\quad with\quad E(w)=0\quad and\quad V(w)=\sigma^ 2W. \] X is an \(n\times q\)-matrix and W is an \(n\times n\) matrix with rank k. W can be written as \(W=FF'\) where F is an \(n\times k\)-matrix. This model is studied using singular value decomposition of F and X. Then it is possible to determine the estimable part, say \(\beta^{(e)}\), of \(\beta\) and to exhibit a linear estimator of \(\beta^{(e)}\). Its properties, including optimality, are proved.