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Yıl 2021, Cilt: 9 Sayı: 4, 185 - 193, 31.12.2021

Melike Yiğit Nesrin Güler Melek Eriş Büyükkaya

https://doi.org/10.36753/mathenot.892258

Öz

Kaynakça

  • [1] Alalouf, I. S., Styan, G. P. H.: Characterizations of estimability in the general linear model. Ann. Stat. 7, 194-200 (1979).
  • [2] Arendacká, B., Puntanen, S.: Further remarks on the connection between fixed linear model and mixed linear model. Stat. Papers. 56 (4), 1235-1247 (2015).
  • [3] Baksalary, J. K., Kala, S.: Linear transformations preserving best linear unbiased estimators in a general Gauss-Markoff model. Ann. Stat. 9, 913-916 (1981).
  • [4] Dong, B., Guo, W., Tian, Y.: On relations between BLUEs under two transformed linear models. J. Multivariate Anal. 131, 279-292 (2014).
  • [5] Drygas, H.: Sufficiency and completeness in the general Gauss-Markov model. Sankhya ̄, Ser A. 45, 88-98 (1983).
  • [6] Goldberger, A. S.: Best linear unbiased prediction in the generalized linear regression model. J. Amer. Statist. Assoc. 57, 369-375 (1962).
  • [7] Güler, N.: On relations between BLUPs under two transformed linear random-effects models. Commun. Statist. Simulation and Computation. (2020). https://doi.org/10.1080/03610918.2020.1757709
  • [8] Harville, D.: Extension of the Gauss-Markov theorem to include the estimation of random effects. Ann. Stat. 4, 384-395 (1976).
  • [9] Haslett, S. J., Puntanen, S.: Equality of BLUEs or BLUPs under two linear models using stochastic restrictions. Stat. Papers. 51 (2), 465-475 (2010).
  • [10] Haslett, S. J., Puntanen, S.: On the equality of the BLUPs under two linear mixed models. Metrika. 74, 381-395 (2011).
  • [11] Isotalo, J., Puntanen, S.: Linear prediction sufficiency for new observations in the general Gauss–Markov model.
  • Commun. Statist. Theory and Methods. 35, 1011-1023 (2006).
  • [12] Liu, X., Wang, Q. W.: Equality of the BLUPs under the mixed linear model when random components and errors are correlated. J. Multivariate Anal. 116, 297-309 (2013).
  • [13] Marsaglia, G., Styan, G. P. H.: Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra. 2, 269-292 (1974).
  • [14] Penrose, R.: Generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51, 406-413 (1955).
  • [15] Puntanen, S., Styan, G. P. H. , Isotalo, J.: Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Springer, Heidelberg (2011).
  • [16] Rao, C. R.: Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J. Multivariate Anal. 3, 276-292 (1973).
  • [17] Sun, Y., Jiang B., Jiang, H.: Computations of predictors/estimators under a linear random–effects model with parameter restrictions. Commun. Statist. Theory and Methods. 48 (14), 3482-3497 (2019).
  • [18] Tian, Y.: The maximal and minimal ranks of some expressions of generalized inverses of matrices. Southeast Asian Bull. Math. 25, 745-755 (2002).
  • [19] Tian, Y.: On equalities for BLUEs under misspecified Gauss-Markov models. Acta Mathematica Sinica. Eng. Ser. 25 (11), 1907-1920 (2009).
  • [20] Tian, Y.: A new derivation of BLUPs under random-effects model. Metrika. 78, 905-918 (2015).
  • [21] Tian, Y.: A matrix handling of predictions under a general linear random-effects model with new observations. Electron. J. Linear Algebra. 29, 30-45 (2015).
  • [22] Tian, Y.: Transformation approaches of linear random-effects models. Stat. Methods Appl. 26 (4), 583-608 (2017).
  • [23] Tian, Y., Cheng, S.: The maximal and minimal ranks of A-BXC with applications. New York J. Math. 9, 345-362 (2003).
  • [24] Tian, Y., Puntanen, S.: On the equivalence of estimations under a general linear model and its transformed models. Linear Algebra Appl. 430, 2622-2641 (2009).
  • [25] Tian, Y., Jiang, B.: An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices. Linear Multilinear Algebra. 64 (12), 2351-2367 (2016).
  • [26] Tian, Y., Jiang B.: Matrix rank/inertia formulas for least-squares solutions with statistical applications. Spec. Matrices. 4 (1), 130-140 (2016).

Some Remarks on the Equalities of Predictors in Linear Mixed Models

Yıl 2021, Cilt: 9 Sayı: 4, 185 - 193, 31.12.2021

Melike Yiğit Nesrin Güler Melek Eriş Büyükkaya

https://doi.org/10.36753/mathenot.892258

Öz


Consider a transformed linear mixed model (TLMM) obtained pre-multiplying a linear mixed model (LMM) M : y = Zα + Rγ + e by a given matrix. This work concerns the problem of the equalities of linear predictors under the considered two LMMs under general assumptions. We characterize the equalities between the best linear unbiased predictors (BLUPs) under the LMM and its TLMM by using various rank formulas of block matrices and elementary matrix operations.

Anahtar Kelimeler

BLUP equalities linear mixed model random vectors transformed model

Kaynakça

  • [1] Alalouf, I. S., Styan, G. P. H.: Characterizations of estimability in the general linear model. Ann. Stat. 7, 194-200 (1979).
  • [2] Arendacká, B., Puntanen, S.: Further remarks on the connection between fixed linear model and mixed linear model. Stat. Papers. 56 (4), 1235-1247 (2015).
  • [3] Baksalary, J. K., Kala, S.: Linear transformations preserving best linear unbiased estimators in a general Gauss-Markoff model. Ann. Stat. 9, 913-916 (1981).
  • [4] Dong, B., Guo, W., Tian, Y.: On relations between BLUEs under two transformed linear models. J. Multivariate Anal. 131, 279-292 (2014).
  • [5] Drygas, H.: Sufficiency and completeness in the general Gauss-Markov model. Sankhya ̄, Ser A. 45, 88-98 (1983).
  • [6] Goldberger, A. S.: Best linear unbiased prediction in the generalized linear regression model. J. Amer. Statist. Assoc. 57, 369-375 (1962).
  • [7] Güler, N.: On relations between BLUPs under two transformed linear random-effects models. Commun. Statist. Simulation and Computation. (2020). https://doi.org/10.1080/03610918.2020.1757709
  • [8] Harville, D.: Extension of the Gauss-Markov theorem to include the estimation of random effects. Ann. Stat. 4, 384-395 (1976).
  • [9] Haslett, S. J., Puntanen, S.: Equality of BLUEs or BLUPs under two linear models using stochastic restrictions. Stat. Papers. 51 (2), 465-475 (2010).
  • [10] Haslett, S. J., Puntanen, S.: On the equality of the BLUPs under two linear mixed models. Metrika. 74, 381-395 (2011).
  • [11] Isotalo, J., Puntanen, S.: Linear prediction sufficiency for new observations in the general Gauss–Markov model.
  • Commun. Statist. Theory and Methods. 35, 1011-1023 (2006).
  • [12] Liu, X., Wang, Q. W.: Equality of the BLUPs under the mixed linear model when random components and errors are correlated. J. Multivariate Anal. 116, 297-309 (2013).
  • [13] Marsaglia, G., Styan, G. P. H.: Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra. 2, 269-292 (1974).
  • [14] Penrose, R.: Generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51, 406-413 (1955).
  • [15] Puntanen, S., Styan, G. P. H. , Isotalo, J.: Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Springer, Heidelberg (2011).
  • [16] Rao, C. R.: Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J. Multivariate Anal. 3, 276-292 (1973).
  • [17] Sun, Y., Jiang B., Jiang, H.: Computations of predictors/estimators under a linear random–effects model with parameter restrictions. Commun. Statist. Theory and Methods. 48 (14), 3482-3497 (2019).
  • [18] Tian, Y.: The maximal and minimal ranks of some expressions of generalized inverses of matrices. Southeast Asian Bull. Math. 25, 745-755 (2002).
  • [19] Tian, Y.: On equalities for BLUEs under misspecified Gauss-Markov models. Acta Mathematica Sinica. Eng. Ser. 25 (11), 1907-1920 (2009).
  • [20] Tian, Y.: A new derivation of BLUPs under random-effects model. Metrika. 78, 905-918 (2015).
  • [21] Tian, Y.: A matrix handling of predictions under a general linear random-effects model with new observations. Electron. J. Linear Algebra. 29, 30-45 (2015).
  • [22] Tian, Y.: Transformation approaches of linear random-effects models. Stat. Methods Appl. 26 (4), 583-608 (2017).
  • [23] Tian, Y., Cheng, S.: The maximal and minimal ranks of A-BXC with applications. New York J. Math. 9, 345-362 (2003).
  • [24] Tian, Y., Puntanen, S.: On the equivalence of estimations under a general linear model and its transformed models. Linear Algebra Appl. 430, 2622-2641 (2009).
  • [25] Tian, Y., Jiang, B.: An algebraic study of BLUPs under two linear random-effects models with correlated covariance matrices. Linear Multilinear Algebra. 64 (12), 2351-2367 (2016).
  • [26] Tian, Y., Jiang B.: Matrix rank/inertia formulas for least-squares solutions with statistical applications. Spec. Matrices. 4 (1), 130-140 (2016).
Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Melike Yiğit 0000-0002-9205-7842

Nesrin Güler 0000-0003-3233-5377

Melek Eriş Büyükkaya 0000-0002-6207-5687

Yayımlanma Tarihi 31 Aralık 2021
Gönderilme Tarihi 6 Mart 2021
Kabul Tarihi 25 Kasım 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 9 Sayı: 4

Kaynak Göster

APA Yiğit, M., Güler, N., & Eriş Büyükkaya, M. (2021). Some Remarks on the Equalities of Predictors in Linear Mixed Models. Mathematical Sciences and Applications E-Notes, 9(4), 185-193. https://doi.org/10.36753/mathenot.892258
AMA Yiğit M, Güler N, Eriş Büyükkaya M. Some Remarks on the Equalities of Predictors in Linear Mixed Models. Math. Sci. Appl. E-Notes. Aralık 2021;9(4):185-193. doi:10.36753/mathenot.892258
Chicago Yiğit, Melike, Nesrin Güler, ve Melek Eriş Büyükkaya. “Some Remarks on the Equalities of Predictors in Linear Mixed Models”. Mathematical Sciences and Applications E-Notes 9, sy. 4 (Aralık 2021): 185-93. https://doi.org/10.36753/mathenot.892258.
EndNote Yiğit M, Güler N, Eriş Büyükkaya M (01 Aralık 2021) Some Remarks on the Equalities of Predictors in Linear Mixed Models. Mathematical Sciences and Applications E-Notes 9 4 185–193.
IEEE M. Yiğit, N. Güler, ve M. Eriş Büyükkaya, “Some Remarks on the Equalities of Predictors in Linear Mixed Models”, Math. Sci. Appl. E-Notes, c. 9, sy. 4, ss. 185–193, 2021, doi: 10.36753/mathenot.892258.
ISNAD Yiğit, Melike vd. “Some Remarks on the Equalities of Predictors in Linear Mixed Models”. Mathematical Sciences and Applications E-Notes 9/4 (Aralık 2021), 185-193. https://doi.org/10.36753/mathenot.892258.
JAMA Yiğit M, Güler N, Eriş Büyükkaya M. Some Remarks on the Equalities of Predictors in Linear Mixed Models. Math. Sci. Appl. E-Notes. 2021;9:185–193.
MLA Yiğit, Melike vd. “Some Remarks on the Equalities of Predictors in Linear Mixed Models”. Mathematical Sciences and Applications E-Notes, c. 9, sy. 4, 2021, ss. 185-93, doi:10.36753/mathenot.892258.
Vancouver Yiğit M, Güler N, Eriş Büyükkaya M. Some Remarks on the Equalities of Predictors in Linear Mixed Models. Math. Sci. Appl. E-Notes. 2021;9(4):185-93.
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