General linear mixed model and signal extraction problem with constraint

作者：Dermoune Azzouz^*; Rahmania Nadji; Wei Tianwen

来源：Journal of Multivariate Analysis, 2012, 105(1): 311-321.

DOI：10.1016/j.jmva.2011.10.007

• Signal extraction
• Generalized inverse
• Asymptotic property
• BLUE
• BLUP
• Auto-regressive model
• Maximum likelihood estimator
• Inverse problem

We consider a noisy observed vector y = x + u is an element of R-n. 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 (beta, F, A(-)), where beta 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 (beta, F, A(-)) defines the solution x = F beta + A(-) v and the general linear mixed model y = F beta + 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 beta is a parameter and (F, A(-)) is given, then we will study the asymptotic behavior of the best linear estimator of beta. If the constraint (beta, F, A(-)) is not known, then we will estimate it using the data y. Some numerical results will be given.

• 出版日期2012-2