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On best linear unbiased estimation and prediction under a constrained linear random-effects model

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  • This paper is concerned with solving some fundamental estimation, prediction, and inference problems on a linear random-effects model with its parameter vector satisfying certain exact linear restrictions. Our work includes deriving analytical formulas for calculating the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of all unknown parameters in the model by way of solving certain constrained quadratic matrix optimization problems, characterizing various mathematical and statistical properties of the predictors and estimators, establishing various fundamental rank and inertia formulas associated with the covariance matrices of predictors and estimators, and presenting necessary and sufficient conditions for several equalities and inequalities of covariance matrices of the predictors and estimators to hold.

    Mathematics Subject Classification: Primary: 62F10; Secondary: 62F30, 62J05.


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  • [1] L. J. EdwardsP. W. StewartK. E. Muller and R. W. Helms, Linear equality constraints in the general linear mixed model, Biometrics, 57 (2001), 1185-1190.  doi: 10.1111/j.0006-341X.2001.01185.x.
    [2] S. GanC. Lu and Y. Tian, Computation and comparison of estimators under different linear random-effects models, Commun. Statist. Simul. Comput., 49 (2020), 1210-1222.  doi: 10.1080/03610918.2018.1493507.
    [3] S. GanY. Sun and Y. Tian, Equivalence of predictors under real and over-parameterized linear models, Commun. Statist. Theor. Meth., 46 (2017), 5368-5383.  doi: 10.1080/03610926.2015.1100742.
    [4] A. S. Goldberger, Best linear unbiased prediction in the generalized linear regression models, J. Amer. Stat. Assoc., 57 (1962), 369-375.  doi: 10.1080/01621459.1962.10480665.
    [5] N. Güler, On relations between BLUPs under two transformed linear random-effects models, Commun. Statist. Simul. Comput., 2020 doi: 10.1080/03610918.2020.1757709.
    [6] N. Güler and M. E. Büyükkaya, Notes on comparison of covariance matrices of BLUPs under linear random-efects model with its two subsample models, Iran. J. Sci. Technol. Trans. A Sci., 43 (2019), 2993-3002.  doi: 10.1007/s40995-019-00785-3.
    [7] N. Güler and M. E. Büyükkaya, Rank and inertia formulas for covariance matrices of BLUPs in general linear mixed models, Comm. Statist. Theory Methods, 50 (2021), 4997-5012.  doi: 10.1080/03610926.2019.1599950.
    [8] N. Güler and M. E. Büyükkaya, Inertia and rank approach in transformed linear mixed models for comparison of BLUPs, Commun. Statist. Theor. Meth., 2021. doi: 10.1080/03610926.2021.1967397.
    [9] D. Harville, Extension of the Gauss–Markov theorem to include the estimation of random effects, Ann. Statist., 4 (1976), 384-395. 
    [10] C. R. Henderson, Best linear unbiased estimation and prediction under a selection model, Biometrics, 31 (1975), 423-447.  doi: 10.2307/2529430.
    [11] J. Hou and B. Jiang, Predictions and estimations under a group of linear models with random coefficients, Comm. Statist. Simulation Comput., 47 (2018), 510-525.  doi: 10.1080/03610918.2017.1283704.
    [12] B. Jiang and Y. Sun, On the equality of estimators under a general partitioned linear model with parameter restrictions, Stat. Papers, 60 (2019), 273-292.  doi: 10.1007/s00362-016-0837-9.
    [13] B. Jiang and Y. Tian, Decomposition approaches of a constrained general linear model with fixed parameters, Electron. J. Linear Algebra, 32 (2017), 232-253.  doi: 10.13001/1081-3810.3428.
    [14] B. Jiang and Y. Tian, On additive decompositions of estimators under a multivariate general linear model and its two submodels, J. Multivariate Anal., 162 (2017), 193-214.  doi: 10.1016/j.jmva.2017.09.007.
    [15] B. Jiang and Y. Tian, On equivalence of predictors/estimators under a multivariate general linear model with augmentation, J. Korean Stat. Soc., 46 (2017), 551-561.  doi: 10.1016/j.jkss.2017.04.001.
    [16] B. JiangY. Tian and X. Zhang, On decompositions of estimators under a general linear model with partial parameter restrictions, Open Math., 15 (2017), 1300-1322.  doi: 10.1515/math-2017-0109.
    [17] H. JiangJ. Qian and Y. Sun, Best linear unbiased predictors and estimators under a pair of constrained seemingly unrelated regression models, Stat. Probab. Lett., 158 (2020), 108669.  doi: 10.1016/j.spl.2019.108669.
    [18] M. Liu, Y. Tian and R. Yuan, Statistical inference of a partitioned linear random-effects model, Commun. Statist. Theor. Meth., 2021. doi: 10.1080/03610926.2021.1926509.
    [19] C. LuY. Sun and Y. Tian, Two competing linear random-effects models and their connections, Statist. Papers, 59 (2018), 1101-1115.  doi: 10.1007/s00362-016-0806-3.
    [20] A. Markiewicz and S. Puntanen, All about the $\perp$ with its applications in the linear statistical models, Open Math., 13 (2015), 33-50.  doi: 10.1515/math-2015-0005.
    [21] A. Markiewicz and S. Puntanen, Further properties of linear prediction sufficiency and the BLUPs in the linear model with new observations, Afr. Stat., 13 (2018), 1511-1530.  doi: 10.16929/as/1511.117.
    [22] G. Marsaglia and G. P. H. Styan, Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra, 2 (1974), 269-292.  doi: 10.1080/03081087408817070.
    [23] C. A. Mcgilchrist and C. W. Aisbett, Restricted BLUP for mixed linear models, Biometr. J., 33 (1991), 131-141.  doi: 10.1002/bimj.4710330202.
    [24] S. K. Mitra, Generalized inverse of matrices and applications to linear models, Handbook of Statistics, (ed. P. K. Krishnaiah), North-Holland Publishing Company, 1 (1980), 471–512.
    [25] M. Möls, Constraints on random effects and mixed linear model predictions, Acta Appl. Math., 79 (2003), 17-23.  doi: 10.1023/A:1025810205448.
    [26] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Phil. Soc., 51 (1955), 406-413.  doi: 10.1017/S0305004100030401.
    [27] S. Puntanen, G. P. H. Styan and J. Isotalo, Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty, Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-10473-2.
    [28] C. R. Rao, Unified theory of linear estimation, Sankhya = a Ser. A, 33 (1971), 371-394. 
    [29] 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.  doi: 10.1016/0047-259X(73)90042-0.
    [30] C. R. Rao, Simultaneous estimation of parameters in different linear models and applications to biometric problems, Biometrics, 31 (1975), 545-554.  doi: 10.2307/2529436.
    [31] C. R. Rao, A lemma on optimization of matrix function and a review of the unified theory of linear estimation, Statistical Data Analysis and Inference, (ed. Y. Dodge), North Holland, (1989), 397–417.
    [32] C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, John Wiley & Sons, Inc., New York-London-Sydney, 1971.
    [33] Y. SunB. Jiang and H. Jiang, Computations of predictors/estimators under a linear random-effects model with parameter restrictions, Commum. Statist. Theor. Meth., 48 (2019), 3482-3497.  doi: 10.1080/03610926.2018.1476714.
    [34] Y. Tian, More on maximal and minimal ranks of Schur complements with applications, Appl. Math. Comput., 152 (2004), 675-692.  doi: 10.1016/S0096-3003(03)00585-X.
    [35] Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433 (2010), 263-296.  doi: 10.1016/j.laa.2010.02.018.
    [36] Y. Tian, A new derivation of BLUPs under random-effects model, Metrika, 78 (2015), 905-918.  doi: 10.1007/s00184-015-0533-0.
    [37] Y. Tian, A matrix handling of predictions of new observations under a general random-effects model, Electron. J. Linear Algebra, 29 (2015), 30-45.  doi: 10.13001/1081-3810.2895.
    [38] Y. Tian, Solutions of a constrained Hermitian matrix-valued function optimization problem with applications, Oper. Matrices, 10 (2016), 967-983.  doi: 10.7153/oam-10-54.
    [39] Y. Tian, Transformation approaches of linear random-effects models, Stat. Methods Appl., 26 (2017), 583-608.  doi: 10.1007/s10260-017-0381-3.
    [40] Y. Tian and B. Jiang, Matrix rank/inertia formulas for least-squares solutions with statistical applications, Spec. Matrices, 4 (2016), 130-140.  doi: 10.1515/spma-2016-0013.
    [41] Y. Tian and B. Jiang, Rank/inertia approaches to weighted least-squares solutions of linear matrix equations, Appl. Math. Comput., 315 (2017), 400-413.  doi: 10.1016/j.amc.2017.07.079.
    [42] Y. Tian and B. Jiang, Quadratic properties of least-squares solutions of linear matrix equations with statistical applications, Comput. Statist., 32 (2017), 1645-1663.  doi: 10.1007/s00180-016-0693-z.
    [43] Y. Tian and P. Xie, Simultaneous optimal predictions under two seemingly unrelated linear random-effects models, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020168.
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