January  2022, 18(1): 561-573. doi: 10.3934/jimo.2020168

Simultaneous optimal predictions under two seemingly unrelated linear random-effects models

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

* Corresponding author: Yongge Tian

Received  May 2020 Revised  September 2020 Published  January 2022 Early access  November 2020

This paper considers simultaneous optimal prediction and estimation problems in the context of linear random-effects models. Assume a pair of seemingly unrelated linear random-effects models (SULREMs) with the random-effects and the error terms correlated. Our aim is to find analytical formulas for calculating best linear unbiased predictors (BLUPs) of all unknown parameters in the two models by means of solving a constrained quadratic matrix optimization problem in the Löwner sense. We also present a variety of theoretical and statistical properties of the BLUPs under the two models.

Citation: 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
References:
[1]

N. K. Bansal and K. J. Miescke, Simultaneous selection and estimation in general linear models, J. Stat. Plann. Inference, 104 (2002), 377-390.  doi: 10.1016/S0378-3758(01)00262-2.

[2]

A. S. Bryk, S. W. Raudenbush and R. T. Congdon, Hierarchical Linear and Nonlinear Modeling with HLM/2L and HLM/3L Programs, Scientific Software International, Chicago, IL, 1996.

[3]

A. Chaturvedi, S. Kesarwani and R. Chandra, Simultaneous prediction based on shrinkage estimator, in: Recent Advances in Linear Models and Related Areas, Essays in Honour of Helge Toutenburg, Springer, 2008, pp. 181–204. doi: 10.1007/978-3-7908-2064-5_10.

[4]

A. ChaturvediA. T. K. Wan and S. P. Singh, Improved multivariate prediction in a general linear model with an unknown error covariance matrix, J. Multivariate Anal., 83 (2002), 166-182.  doi: 10.1006/jmva.2001.2042.

[5]

M. Dube and V. Manocha, Simultaneous prediction in restricted regression models, J. Appl. Statist. Sci., 11 (2002), 277-288. 

[6]

B. Effron and C. Morris, Combining possibly related estimation problems (with discussion), J. Roy. Stat. Soc. B, 35 (1973), 379–421. https://www.jstor.org/stable/2985106

[7]

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.

[8] A. Gelman and J. Hill, Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press, 2007. 
[9]

H. Goldstein and J. D. Leeuw, Handbook of Multilevel Analysis, Springer New York, 2008.

[10]

C. A. Gotway and N. Cressie, Improved multivariate prediction under a general linear model, J. Multivariate Anal., 45 (1993), 56-72.  doi: 10.1006/jmva.1993.1026.

[11]

N. Güler and M. E. Büyükkaya, Rank and inertia formulas for covariance matrices of BLUPs in general linear mixed models, Commun. Statist. Theor. Meth., 2020. doi: 10.1080/03610926.2019.1599950.

[12]

S. J. Haslett and S. Puntanen, Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Stat. Papers, 51 (2010), 465-475.  doi: 10.1007/s00362-009-0219-7.

[13]

S. J. Haslett and S. Puntanen, A note on the equality of the BLUPs for new observations under two linear models, Acta Comm. Univ. Tartu. Math., 14 (2010), 27-33. 

[14]

S. J. Haslett and S. Puntanen, On the equality of the BLUPs under two linear mixed models, Metrika, 74 (2011), 381-395.  doi: 10.1007/s00184-010-0308-6.

[15]

J. Hou and B. Jiang, Predictions and estimations under a group of linear models with random coefficients, Comm. Statist. Simul. Comput., 47 (2018), 510-525.  doi: 10.1080/03610918.2017.1283704.

[16]

H. Jiang, J. 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.

[17]

C. LuY. Sun and Y. Tian, A comparison between two competing fixed parameter constrained general linear models with new regressors, Statistics, 52 (2018), 769-781.  doi: 10.1080/02331888.2018.1469021.

[18]

C. LuY. Sun and Y. Tian, Two competing linear random-effects models and their connections, Stat. Papers, 59 (2018), 1101-1115.  doi: 10.1007/s00362-016-0806-3.

[19]

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.

[20]

S. K. Mitra, Generalized inverse of matrices and applications to linear models, in: Handbook of Statistics, P.K. Krishnaiah, ed., Vol. 1, North-Holland, pp. 471–512, 1980.

[21]

R. Penrose, A generalized inverse for matrices, Proc. Cambridge Phil. Soc., 51 (1955), 406-413.  doi: 10.1017/S0305004100030401.

[22]

S. Puntanen, G. P. H. Styan and J. Isotalo, Matrix Tricks for Linear Statistical Models, Our Personal Top Twenty, Springer, Berlin, 2011. doi: 10.1007/978-3-642-10473-2.

[23]

C. R. Rao, Unified theory of linear estimation, Sankhyā, Ser. A, 33 (1971), 371-394. 

[24]

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.

[25]

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.

[26]

C. R. Rao, A lemma on optimization of matrix function and a review of the unified theory of linear estimation, in: Statistical Data Analysis and Inference, Y. Dodge (ed.), North Holland, 1989, pp. 397–417.

[27]

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971.,

[28]

C. R. Rao, H. Toutenburg, Shalabh and C. Heumann, Linear Models and Generalizations: Least Squares and Alternatives, 3rd edition, Springer, Berlin, 2008.

[29]

S. W. Raudenbush and A. S. Bryk, Hierarchical Linear Models: Applications and Data Analysis Methods, 2nd edition, Sage, Thousand Oaks, 2002.

[30]

Sh alabh, Performance of Stein-rule procedure for simultaneous prediction of actual and average values of study variable in linear regression models, Bull. Internat. Stat. Instit., 56 (1995), 1375-1390. 

[31]

Y. SunB. Jiang and H. Jiang, Computations of predictors/estimators under a linear random-effects model with parameter restrictions, Comm. Statist. Theory Meth., 48 (2019), 3482-3497.  doi: 10.1080/03610926.2018.1476714.

[32]

Y. Sun, H. Jiang and Y. Tian, A prediction analysis in a constrained multivariate general linear model with future observations, Comm. Statist. Theory Meth., 2020. doi: 10.1080/03610926.2019.1634819.

[33]

Y. Tian, A new derivation of BLUPs under random-effects model, Metrika, 78 (2015), 905-918.  doi: 10.1007/s00184-015-0533-0.

[34]

Y. Tian, A matrix handling of predictions under a general linear random-effects model with new observations, Electron. J. Linear Algebra, 29 (2015), 30-45.  doi: 10.13001/1081-3810.2895.

[35]

Y. Tian, Transformation approaches of linear random-effects models, Statist. Meth. Appl., 26 (2017), 583-608.  doi: 10.1007/s10260-017-0381-3.

[36]

Y. Tian and B. Jiang, An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices, Linear Multilinear Algebra, 64 (2016), 2351-2367.  doi: 10.1080/03081087.2016.1155533.

[37]

Y. Tian and J. Wang, Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model, Commun. Statist. Theory Meth., 49 (2020), 1201-1216.  doi: 10.1080/03610926.2018.1554138.

[38]

H. Toutenburg, Prior Information in Linear Models., Wiley, New York, 1982.

[39]

H. Toutenburg and Sh alabh, Predictive performance of the methods of restricted and mixed regression estimators, Biometr. J., 38 (1996), 951-959. 

[40]

H. Toutenburg and Sh alabh, Improved prediction in linear regression model with stochastic linear constraints, Biometr. J., 42 (2000), 71-86.  doi: 10.1002/(SICI)1521-4036(200001)42:1<71::AID-BIMJ71>3.0.CO;2-H.

show all references

References:
[1]

N. K. Bansal and K. J. Miescke, Simultaneous selection and estimation in general linear models, J. Stat. Plann. Inference, 104 (2002), 377-390.  doi: 10.1016/S0378-3758(01)00262-2.

[2]

A. S. Bryk, S. W. Raudenbush and R. T. Congdon, Hierarchical Linear and Nonlinear Modeling with HLM/2L and HLM/3L Programs, Scientific Software International, Chicago, IL, 1996.

[3]

A. Chaturvedi, S. Kesarwani and R. Chandra, Simultaneous prediction based on shrinkage estimator, in: Recent Advances in Linear Models and Related Areas, Essays in Honour of Helge Toutenburg, Springer, 2008, pp. 181–204. doi: 10.1007/978-3-7908-2064-5_10.

[4]

A. ChaturvediA. T. K. Wan and S. P. Singh, Improved multivariate prediction in a general linear model with an unknown error covariance matrix, J. Multivariate Anal., 83 (2002), 166-182.  doi: 10.1006/jmva.2001.2042.

[5]

M. Dube and V. Manocha, Simultaneous prediction in restricted regression models, J. Appl. Statist. Sci., 11 (2002), 277-288. 

[6]

B. Effron and C. Morris, Combining possibly related estimation problems (with discussion), J. Roy. Stat. Soc. B, 35 (1973), 379–421. https://www.jstor.org/stable/2985106

[7]

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.

[8] A. Gelman and J. Hill, Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press, 2007. 
[9]

H. Goldstein and J. D. Leeuw, Handbook of Multilevel Analysis, Springer New York, 2008.

[10]

C. A. Gotway and N. Cressie, Improved multivariate prediction under a general linear model, J. Multivariate Anal., 45 (1993), 56-72.  doi: 10.1006/jmva.1993.1026.

[11]

N. Güler and M. E. Büyükkaya, Rank and inertia formulas for covariance matrices of BLUPs in general linear mixed models, Commun. Statist. Theor. Meth., 2020. doi: 10.1080/03610926.2019.1599950.

[12]

S. J. Haslett and S. Puntanen, Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Stat. Papers, 51 (2010), 465-475.  doi: 10.1007/s00362-009-0219-7.

[13]

S. J. Haslett and S. Puntanen, A note on the equality of the BLUPs for new observations under two linear models, Acta Comm. Univ. Tartu. Math., 14 (2010), 27-33. 

[14]

S. J. Haslett and S. Puntanen, On the equality of the BLUPs under two linear mixed models, Metrika, 74 (2011), 381-395.  doi: 10.1007/s00184-010-0308-6.

[15]

J. Hou and B. Jiang, Predictions and estimations under a group of linear models with random coefficients, Comm. Statist. Simul. Comput., 47 (2018), 510-525.  doi: 10.1080/03610918.2017.1283704.

[16]

H. Jiang, J. 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.

[17]

C. LuY. Sun and Y. Tian, A comparison between two competing fixed parameter constrained general linear models with new regressors, Statistics, 52 (2018), 769-781.  doi: 10.1080/02331888.2018.1469021.

[18]

C. LuY. Sun and Y. Tian, Two competing linear random-effects models and their connections, Stat. Papers, 59 (2018), 1101-1115.  doi: 10.1007/s00362-016-0806-3.

[19]

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.

[20]

S. K. Mitra, Generalized inverse of matrices and applications to linear models, in: Handbook of Statistics, P.K. Krishnaiah, ed., Vol. 1, North-Holland, pp. 471–512, 1980.

[21]

R. Penrose, A generalized inverse for matrices, Proc. Cambridge Phil. Soc., 51 (1955), 406-413.  doi: 10.1017/S0305004100030401.

[22]

S. Puntanen, G. P. H. Styan and J. Isotalo, Matrix Tricks for Linear Statistical Models, Our Personal Top Twenty, Springer, Berlin, 2011. doi: 10.1007/978-3-642-10473-2.

[23]

C. R. Rao, Unified theory of linear estimation, Sankhyā, Ser. A, 33 (1971), 371-394. 

[24]

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.

[25]

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.

[26]

C. R. Rao, A lemma on optimization of matrix function and a review of the unified theory of linear estimation, in: Statistical Data Analysis and Inference, Y. Dodge (ed.), North Holland, 1989, pp. 397–417.

[27]

C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971.,

[28]

C. R. Rao, H. Toutenburg, Shalabh and C. Heumann, Linear Models and Generalizations: Least Squares and Alternatives, 3rd edition, Springer, Berlin, 2008.

[29]

S. W. Raudenbush and A. S. Bryk, Hierarchical Linear Models: Applications and Data Analysis Methods, 2nd edition, Sage, Thousand Oaks, 2002.

[30]

Sh alabh, Performance of Stein-rule procedure for simultaneous prediction of actual and average values of study variable in linear regression models, Bull. Internat. Stat. Instit., 56 (1995), 1375-1390. 

[31]

Y. SunB. Jiang and H. Jiang, Computations of predictors/estimators under a linear random-effects model with parameter restrictions, Comm. Statist. Theory Meth., 48 (2019), 3482-3497.  doi: 10.1080/03610926.2018.1476714.

[32]

Y. Sun, H. Jiang and Y. Tian, A prediction analysis in a constrained multivariate general linear model with future observations, Comm. Statist. Theory Meth., 2020. doi: 10.1080/03610926.2019.1634819.

[33]

Y. Tian, A new derivation of BLUPs under random-effects model, Metrika, 78 (2015), 905-918.  doi: 10.1007/s00184-015-0533-0.

[34]

Y. Tian, A matrix handling of predictions under a general linear random-effects model with new observations, Electron. J. Linear Algebra, 29 (2015), 30-45.  doi: 10.13001/1081-3810.2895.

[35]

Y. Tian, Transformation approaches of linear random-effects models, Statist. Meth. Appl., 26 (2017), 583-608.  doi: 10.1007/s10260-017-0381-3.

[36]

Y. Tian and B. Jiang, An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices, Linear Multilinear Algebra, 64 (2016), 2351-2367.  doi: 10.1080/03081087.2016.1155533.

[37]

Y. Tian and J. Wang, Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model, Commun. Statist. Theory Meth., 49 (2020), 1201-1216.  doi: 10.1080/03610926.2018.1554138.

[38]

H. Toutenburg, Prior Information in Linear Models., Wiley, New York, 1982.

[39]

H. Toutenburg and Sh alabh, Predictive performance of the methods of restricted and mixed regression estimators, Biometr. J., 38 (1996), 951-959. 

[40]

H. Toutenburg and Sh alabh, Improved prediction in linear regression model with stochastic linear constraints, Biometr. J., 42 (2000), 71-86.  doi: 10.1002/(SICI)1521-4036(200001)42:1<71::AID-BIMJ71>3.0.CO;2-H.

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