General linear mixed model and signal extraction problem with constraint

We consider a noisy observed vector y=x+u∈Rn. The unobserved vector x is a solution of a non-invertible linear system Ax=v, where v is a forcing term. A unique solution of the system is obtained by considering additional constraint on the vector x. This constraint is defined by a triple (β,F,A−), where β is a vector, F denotes a matrix whose range is equal to N(A) (the null space of A) and A− is a generalized inverse of A. Each triple (β,F,A−) defines the solution x=Fβ+A−v and the general linear mixed model y=Fβ+A−v+u. Given the covariance matrices of u and v, we will prove that the best linear unbiased predictor of x knowing y depends only on A. If β is a parameter and (F,A−) is given, then we will study the asymptotic behavior of the best linear estimator of β. If the constraint (β,F,A−) is not known, then we will estimate it using the data y. Some numerical results will be given.