On the calculation of minimum variance estimators for unobservable dependent variables (Q1315452)

The author considers the problem to find optimal values for the variables \(x_ 1,x_ 2,\dots,x_ k\) minimizing \(S(X) = \text{trace}[(R + PX)K(R + PX)']\), where \(X\) denotes the diagonal matrix with nonzero diagonal elements \(X_{jj} = x_ j\), \(P\) and \(R\) are data matrices and \(K\) is a (positive semidefinite) matrix. He gives the minimum variance estimator and proves its uniqueness by determining necessary and sufficient conditions. Numerical aspects of the method are also briefly explored. This type of problems arises from an attempt to perform a regression with an unobservable dependent variable.