Article
| Title: | Some remarks on comparison of predictors in seemingly unrelated linear mixed models (English) |
| Author: | Güler, Nesrin |
| Author: | Eriş Büyükkaya, Melek |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 67 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 525-542 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | In this paper, we consider a comparison problem of predictors in the context of linear mixed models. In particular, we assume a set of $m$ different seemingly unrelated linear mixed models (SULMMs) allowing correlations among random vectors across the models. Our aim is to establish a variety of equalities and inequalities for comparing covariance matrices of the best linear unbiased predictors (BLUPs) of joint unknown vectors under SULMMs and their combined model. We use the matrix rank and inertia method for establishing equalities and inequalities. We also give an extensive approach for seemingly unrelated regression models (SURMs) by applying the results obtained for SULMMs to SURMs. (English) |
| Keyword: | BLUP |
| Keyword: | covariance matrix |
| Keyword: | inertia |
| Keyword: | OLSP |
| Keyword: | rank |
| Keyword: | SULMM |
| Keyword: | SURM |
| MSC: | 15A03 |
| MSC: | 62H12 |
| MSC: | 62J05 |
| idZBL: | Zbl 07584084 |
| idMR: | MR4444791 |
| DOI: | 10.21136/AM.2021.0366-20 |
| . | |
| Date available: | 2022-06-28T13:23:59Z |
| Last updated: | 2022-12-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150441 |
| . | |
| Reference: | [1] Alalouf, I. S., Styan, G. P. H.: Characterizations of estimability in the general linear model.Ann. Stat. 7 (1979), 194-200. Zbl 0398.62053, MR 0515693, 10.1214/aos/1176344564 |
| Reference: | [2] Arendacká, B., Puntanen, S.: Further remarks on the connection between fixed linear model and mixed linear model.Stat. Pap. 56 (2015), 1235-1247. Zbl 1326.62150, MR 3416897, 10.1007/s00362-014-0634-2 |
| Reference: | [3] Baksalary, J. K., Kala, R.: On the prediction problem in the seemingly unrelated regression equations model.Math. Operationsforsch. Stat., Ser. Stat. 10 (1979), 203-208. Zbl 0417.62053, MR 0544566, 10.1080/02331887908801479 |
| Reference: | [4] Bartels, R., Fiebig, D. G.: A simple characterization of seemingly unrelated regressions models in which OLS is BLUE.Am. Stat. 45 (1991), 137-140. MR 1136119, 10.2307/2684378 |
| Reference: | [5] Brown, H., Prescott, R.: Applied Mixed Models in Medicine.Statistics in Practice. John Wiley & Sons, Hoboken (2006). Zbl 1099.62125, 10.1002/0470023589 |
| Reference: | [6] Demidenko, E.: Mixed Models: Theory and Applications.Wiley Series in Probability and Statistics. John Wiley & Sons, New York (2004). Zbl 1055.62086, MR 2077875, 10.1002/0471728438 |
| Reference: | [7] Dwivedi, T. D., Srivastava, V. K.: Optimality of least squares in the seemingly unrelated regression equation model.J. Econom. 7 (1978), 391-395. Zbl 0389.62050, MR 0547229, 10.1016/0304-4076(78)90062-3 |
| Reference: | [8] Goldberger, A. S.: Best linear unbiased prediction in the generalized linear regression model.J. Am. Stat. Assoc. 57 (1962), 369-375. Zbl 0124.35502, MR 0143295, 10.2307/2281645 |
| Reference: | [9] Gong, L.: Establishing equalities of OLSEs and BLUEs under seemingly unrelated regression models.J. Stat. Theory Pract. 13 (2019), Article ID 5, 10 pages. Zbl 1426.62199, MR 3917080, 10.1007/s42519-018-0015-6 |
| Reference: | [10] Güler, N.: On relations between BLUPs under two transformed linear random-effects models.(to appear) in Commun. Stat., Simulation Comput. 10.1080/03610918.2020.1757709 |
| Reference: | [11] Güler, N., Büyükkaya, M. E.: Notes on comparison of covariance matrices of BLUPs under linear random-effects model with its two subsample models.Iran. J. Sci. Technol., Trans. A, Sci. 43 (2019), 2993-3002. MR 4035916, 10.1007/s40995-019-00785-3 |
| Reference: | [12] Güler, N., Büyükkaya, M. E.: Rank and inertia formulas for covariance matrices of BLUPs in general linear mixed models.Commun. Stat., Theory Methods 50 (2021), 4997-5012. MR 4316598, 10.1080/03610926.2019.1599950 |
| Reference: | [13] Haslett, S. J., Puntanen, S.: On the equality of the BLUPs under two linear mixed models.Metrika 74 (2011), 381-395. Zbl 1226.62066, MR 2835620, 10.1007/s00184-010-0308-6 |
| Reference: | [14] Haslett, S. J., Puntanen, S., Arendacká, B.: The link between the mixed and fixed linear models revisited.Stat. Pap. 56 (2015), 849-861. Zbl 1317.62058, MR 3369433, 10.1007/s00362-014-0611-9 |
| Reference: | [15] Hou, J., Zhao, Y.: Some remarks on a pair of seemingly unrelated regression models.Open Math. 17 (2019), 979-989. Zbl 1428.62229, MR 3998221, 10.1515/math-2019-0077 |
| Reference: | [16] Jiang, J.: Linear and Generalized Linear Mixed Models and Their Applications.Springer Series in Statistics. Springer, New York (2007). Zbl 1152.62040, MR 2308058, 10.1007/978-0-387-47946-0 |
| Reference: | [17] Jiang, H., Qian, J., Sun, Y.: Best linear unbiased predictors and estimators under a pair of constrained seemingly unrelated regression models.Stat. Probab. Lett. 158 (2020), Article ID 108669, 7 pages. Zbl 1434.62096, MR 4027766, 10.1016/j.spl.2019.108669 |
| Reference: | [18] Liu, X.-Q., Rong, J.-Y., Liu, X.-Y.: Best linear unbiased prediction for linear combinations in general mixed linear models.J. Multivariate Anal. 99 (2008), 1503-1517. Zbl 1144.62047, MR 2444809, 10.1016/j.jmva.2008.01.004 |
| Reference: | [19] 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 (2013), 297-309. Zbl 1277.62172, MR 3049906, 10.1016/j.jmva.2012.12.006 |
| Reference: | [20] Puntanen, S., Styan, G. P. H., Isotalo, J.: Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty.Springer, Berlin (2011). Zbl 1291.62014, MR 3013662, 10.1007/978-3-642-10473-2 |
| Reference: | [21] Rao, C. R.: Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix.J. Multivariate Anal. 3 (1973), 276-292. Zbl 0276.62068, MR 0341765, 10.1016/0047-259X(73)90042-0 |
| Reference: | [22] Searle, S. R.: The matrix handling of BLUE and BLUP in the mixed linear model.Linear Algebra Appl. 264 (1997), 291-311. Zbl 0889.62059, MR 1465872, 10.1016/S0024-3795(96)00400-4 |
| Reference: | [23] Srivastava, V. K., Giles, D. E. A.: Seemingly Unrelated Regression Equations Models: Estimation and Inference.Statistics: Textbooks and Monographs 80. Marcel Dekker, New York (1987). Zbl 0638.62108, MR 0930104, 10.1201/9781003065654 |
| Reference: | [24] Sun, Y., Ke, R., Tian, Y.: Some overall properties of seemingly unrelated regression models.AStA, Adv. Stat. Anal. 98 (2014), 103-120. Zbl 1443.62206, MR 3254024, 10.1007/s10182-013-0212-2 |
| Reference: | [25] Tian, Y.: Equalities and inequalities for inertias of Hermitian matrices with applications.Linear Algebra Appl. 433 (2010), 263-296. Zbl 1205.15033, MR 2645083, 10.1016/j.laa.2010.02.018 |
| Reference: | [26] Tian, Y.: Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method.Nonlinear Anal., Theory Methods Appl., Ser. A 75 (2012), 717-734. Zbl 1236.65070, MR 2847451, 10.1016/j.na.2011.09.003 |
| Reference: | [27] Tian, Y.: A new derivation of BLUPs under random-effects model.Metrika 78 (2015), 905-918. Zbl 1329.62264, MR 3407588, 10.1007/s00184-015-0533-0 |
| Reference: | [28] Tian, Y.: Matrix rank and inertia formulas in the analysis of general linear models.Open Math. 15 (2017), 126-150. Zbl 1359.15003, MR 3619144, 10.1515/math-2017-0013 |
| Reference: | [29] Tian, Y.: Some equalities and inequalities for covariance matrices of estimators under linear model.Stat. Pap. 58 (2017), 467-484. Zbl 1365.62207, MR 3649498, 10.1007/s00362-015-0707-x |
| Reference: | [30] Tian, Y., Guo, W.: On comparison of dispersion matrices of estimators under a constrained linear model.Stat. Methods Appl. 25 (2016), 623-649. Zbl 1392.62209, MR 3572450, 10.1007/s10260-016-0350-2 |
| Reference: | [31] Tian, Y., Jiang, B.: Matrix rank/inertia formulas for least-squares solutions with statistical applications.Spec. Matrices 4 (2016), 130-140. Zbl 1333.15006, MR 3459007, 10.1515/spma-2016-0013 |
| Reference: | [32] Tian, Y., Wang, J.: Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model.Commun. Stat., Theory Methods 49 (2020), 1201-1216. MR 4048647, 10.1080/03610926.2018.1554138 |
| Reference: | [33] Tian, Y., Xie, P.: Simultaneous optimal predictions under two seemingly unrelated linear random-effects models.(to appear) in J. Ind. Manag. Optim. MR 4357259, 10.3934/jimo.2020168 |
| Reference: | [34] Verbeke, G., Molenberghs, G.: Linear Mixed Models for Longitudinal Data.Springer Series in Statistics. Springer, New York (2000). Zbl 0956.62055, MR 1880596, 10.1007/978-1-4419-0300-6 |
| Reference: | [35] Wang, Q.-W., Liu, X.: The equalities of BLUPs for linear combinations under two general linear mixed models.Commun. Stat., Theory Methods 42 (2013), 3528-3543. Zbl 06232639, MR 3170949, 10.1080/03610926.2011.633730 |
| Reference: | [36] Zellner, A.: An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias.J. Am. Stat. Assoc. 57 (1962), 348-368. Zbl 0113.34902, MR 0139235, 10.2307/2281644 |
| Reference: | [37] Zellner, A., Huang, D. S.: Further properties of efficient estimators for seemingly unrelated regression equations.Int. Econ. Rev. 3 (1962), 300-313. Zbl 0156.40001, MR 0157439, 10.2307/2525396 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
