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General linear mixed model and signal extraction problem with constraint

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  • Dermoune, Azzouz
  • Rahmania, Nadji
  • Wei, Tianwen

Abstract

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.

Suggested Citation

  • Dermoune, Azzouz & Rahmania, Nadji & Wei, Tianwen, 2012. "General linear mixed model and signal extraction problem with constraint," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 311-321.
    • Handle: RePEc:eee:jmvana:v:105:y:2012:i:1:p:311-321
    DOI: 10.1016/j.jmva.2011.10.007
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    References listed on IDEAS

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    1. Hodrick, Robert J & Prescott, Edward C, 1997. "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 29(1), pages 1-16, February.
    2. Drygas, Hilmar, 1985. "Linear sufficiency and some applications in multilinear estimation," Journal of Multivariate Analysis, Elsevier, vol. 16(1), pages 71-84, February.
    3. Liu, Xu-Qing & Rong, Jian-Ying & Liu, Xiu-Ying, 2008. "Best linear unbiased prediction for linear combinations in general mixed linear models," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1503-1517, September.
    4. Dermoune Azzouz & Djehiche Boualem & Rahmania Nadji, 2009. "Multivariate Extension of the Hodrick-Prescott Filter-Optimality and Characterization," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 13(3), pages 1-35, May.
    5. Schlicht, Ekkehart, 2004. "Estimating the Smoothing Parameter in the So-Called Hodrick-Prescott Filter," Discussion Papers in Economics 304, University of Munich, Department of Economics.
    6. Proietti, Tommaso, 2007. "Signal extraction and filtering by linear semiparametric methods," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 935-958, October.
    7. Lynn LaMotte, 2007. "A direct derivation of the REML likelihood function," Statistical Papers, Springer, vol. 48(2), pages 321-327, April.
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    Cited by:

    1. Dermoune, Azzouz & Preda, Cristian, 2017. "Parametrizations, fixed and random effects," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 162-176.

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