On Predictors and Estimators under a Constrained Partitioned Linear Model and its Reduced Models

Year 2022, , 192 - 200, 23.09.2022

Melek Eriş Büyükkaya , Nesrin Güler

https://doi.org/10.33401/fujma.1134600

Cited By: 1

Abstract

In this study, we consider a partitioned linear model with linear partial parameter constrains, known as a constrained partitioned linear model (CPLM), and its reduced models. A group of formulas on best linear unbiased predictors (BLUPs) and best linear unbiased estimators (BLUEs) in CPLM is derived via some quadratic matrix optimization methods, and further many basic properties of the predictors and estimators are established under some general assumptions. Our main purpose is to derive various inequalities and equalities for the comparison of covariance matrices of BLUPs and BLUEs under CPLM and its reduced models.

Keywords

BLUP, Correctly-reduced models, Covariance matrix, Rank

References

• [1] P. Bhimasankaram, R. SahaRay, On a partitioned linear model and some associated reduced models, Linear Algebra Appl., 264 (1997), 329–339.
• [2] J. Grob, S. Puntanen, Estimation under a general partitioned linear model, Linear Algebra Appl., 321 (2000), 131–144.
• [3] C. R. Rao, Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix, J. Multivariate Anal., 3 (1973), 276–292.
• [4] J. K. Baksalary, P. R. Pordzik, Implied linear restrictions in the general Gauss–Markov model, J. Statist. Plann. Inference, 30 (1992), 237–248.
• [5] A. R. Gallant, T. M. Gerig, Computations for constrained linear models, Econometrics, 12 (1) (1980), 59–84.
• [6] S. J. Haslett, S. Puntanen, Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Stat. Papers, 51 (2010), 465–475.
• [7] H. Jiang, J. Qian, Y. Sun, Best linear unbiased predictors and estimators under a pair of constrained seemingly unrelated regression models, Stat. Probab. Lett., 158 (2020), 108669.
• [8] B. Jiang, Y. Sun, On the equality of estimators under a general partitioned linear model with parameter restrictions, Stat. Papers, 60 (2019), 273–292.
• [9] B. Jiang, Y. Tian, Decomposition approaches of a constrained general linear model with fixed parameters, Electron. J. Linear Algebra, 32 (2017), 232–253.
• [10] B. Jiang, Y. Tian, X. Zhang, On decompositions of estimators under a general linear model with partial parameter restrictions, Open Math., 15 (2017), 1300–1322.
• [11] M. Liu, Y. Tian, R. Yuan, Statistical inference of a partitioned linear random-effects model, Commun. Statist. Theory Methods, (2021), doi: 10.1080/03610926.2021.1926509.
• [12] R. Ma, Y. Tian, A matrix approach to a general partitioned linear model with partial parameter restrictions, Linear Multilinear Algebra, (2020), doi: 10.1080/03081087.2020.1804521.
• [13] Y. Tian, On equalities of estimations of parametric functions under a general linear model and its restricted models, Metrika, 72 (3) (2010b), 313–330.
• [14] Y. Tian, W. Guo, On comparison of dispersion matrices of estimators under a constrained linear model, Stat. Methods Appl., 25 (4) (2016), 623–649.
• [15] Y. Tian, B. Jiang, Equalities for estimators of partial parameters under linear model with restrictions, J. Multivariate Anal., 143 (2016), 299–313.
• [16] Y. Tian, J. Wang, Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model, Commun. Statist. Theory Methods, 49 (2020), 1201–1216.
• [17] H. J. Werner, C. Yapar, On equality constrained generalized least squares selections in the general possibly singular Gauss–Markov model: A projector theoretical approach, Linear Algebra Appl., 237/238 (1996), 359–393.
• [18] X. Zhang, Y. Tian, On decompositions of BLUEs under a partitioned linear model with restrictions, Stat. Papers, 57 (2016), 345–364.
• [19] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010a), 263–296.
• [20] Y. Tian, Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method, Nonlinear Anal., 75 (2012), 717–734.
• [21] I. S. Alalouf, G. P. H. Styan, Characterizations of estimability in the general linear model, Ann. Stat., 7 (1979), 194–200.
• [22] A. S. Goldberger, Best linear unbiased prediction in the generalized linear regression model, J. Amer. Statist. Assoc., 57 (1962), 369–375.
• [23] S. Puntanen, G. P. H. Styan, J. Isotalo, Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty, Springer, Heidelberg, 2011.
• [24] N. Güler, M. E. Büyükkkaya, Inertia and rank approach in transformed linear mixed models for comparison of BLUPs, Commun. Statist. Theory Methods, (2021), doi: 10.1080/03610926.2021.1967397.
• [25] Y. Tian, A new derivation of BLUPs under random-effects model, Metrika, 78 (2015), 905–918.
• [26] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51 (1955), 406–413.
• [27] Y. Tian, On properties of BLUEs under general linear regression models, J. Statist. Plann. Inference, 143 (2013), 771–782.