doi: 10.3934/jimo.2021209
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

a. 

College of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, Shandong, China

b. 

College of Business and Economics, Shanghai Business School, Shanghai, China

* Corresponding author

Received  May 2021 Revised  September 2021 Early access December 2021

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.

Citation: Bo Jiang, Yongge Tian. On best linear unbiased estimation and prediction under a constrained linear random-effects model. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021209
References:
[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.

show all references

References:
[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.

[1]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial and Management Optimization, 2022, 18 (1) : 561-573. doi: 10.3934/jimo.2020168

[2]

Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289

[3]

Seung-Yeal Ha, Myeongju Kang, Hansol Park. Collective behaviors of the Lohe Hermitian sphere model with inertia. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2613-2641. doi: 10.3934/cpaa.2021046

[4]

Qiang Zhang, Ping Chen. Multidimensional balanced credibility model with time effect and two level random common effects. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1311-1328. doi: 10.3934/jimo.2019004

[5]

John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019

[6]

Ciprian D. Coman. Dissipative effects in piecewise linear dynamics. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 163-177. doi: 10.3934/dcdsb.2003.3.163

[7]

Gabriel Montes-Rojas, Pedro Elosegui. Network ANOVA random effects models for node attributes. Journal of Dynamics and Games, 2020, 7 (3) : 239-252. doi: 10.3934/jdg.2020017

[8]

Monica De Angelis, Gaetano Fiore. Diffusion effects in a superconductive model. Communications on Pure and Applied Analysis, 2014, 13 (1) : 217-223. doi: 10.3934/cpaa.2014.13.217

[9]

Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002

[10]

Marie Turčičová, Jan Mandel, Kryštof Eben. Score matching filters for Gaussian Markov random fields with a linear model of the precision matrix. Foundations of Data Science, 2021, 3 (4) : 793-824. doi: 10.3934/fods.2021030

[11]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems and Imaging, 2021, 15 (3) : 519-537. doi: 10.3934/ipi.2021003

[12]

Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks and Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943

[13]

Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507

[14]

Hamid Maarouf. Local Kalman rank condition for linear time varying systems. Mathematical Control and Related Fields, 2022, 12 (2) : 433-446. doi: 10.3934/mcrf.2021029

[15]

Tomás Caraballo, Renato Colucci, Xiaoying Han. Semi-Kolmogorov models for predation with indirect effects in random environments. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2129-2143. doi: 10.3934/dcdsb.2016040

[16]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[17]

Nguyen Dinh Cong, Doan Thai Son. On integral separation of bounded linear random differential equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 995-1007. doi: 10.3934/dcdss.2016038

[18]

Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661

[19]

Jonathan Bennett. A trilinear restriction problem for the paraboloid in R^3. Electronic Research Announcements, 2004, 10: 97-102.

[20]

Marion Weedermann. Analysis of a model for the effects of an external toxin on anaerobic digestion. Mathematical Biosciences & Engineering, 2012, 9 (2) : 445-459. doi: 10.3934/mbe.2012.9.445

2020 Impact Factor: 1.801

Article outline

[Back to Top]