General linear mixed model and signal extraction problem with constraint

We consider a noisy observed vector y = x + u ∈ R n . The unobserved vector x is a solution of a non-invertible linear system A x = 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 = F β + A − v and the general linear mixed model y = F β + 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.