Joining Iso-Structured Models with Commutative Orthogonal Block Structure

bc2485ed-b8ac-4cf4-9052-2944fcf5dfdc20230727094958436wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=4051919202291920222210.37394/23206.2023.22https://wseas.com/journals/mathematics/2023.phpJoining Iso-Structured Models with Commutative Orthogonal Block StructureCarlaSantosPolytechnic Institute of Beja, Beja, PORTUGALCristinaDiasPolytechnic Institute of Portalegre, Portalegre, PORTUGALCéliaNunesDepartment of Mathematics and Center of Mathematics and Applications, University of Beira Interior, Covilhã, PORTUGALJoão TiagoMexiaDepartment of Mathematics, SST, New University of Lisbon, Caparica, PORTUGALIn this work, we focus on a special class of mixed models, named models with commutative orthogonal block structure (COBS), whose covariance matrix is a linear combination of known pairwise orthogonal projection matrices that add to the identity matrix, and for which the orthogonal projection matrix on the space spanned by the mean vector commutes with the covariance matrix. The COBS have least squares estimators giving the best linear unbiased estimators for estimable vectors. Our approach to COBS relies on their algebraic structure, based on commutative Jordan algebras of symmetric matrices, which proves to be advantageous as it leads to important results in the estimation. Specifically, we are interested in iso-structured COBS, applying to them the operation of models joining. We show that joining iso-structured COBS gives COBS and that the estimators for the joint model may be obtained from those for the individual models.7272023727202357758564https://wseas.com/journals/mathematics/2023/b305106-031(2023).pdf10.37394/23206.2023.22.64https://wseas.com/journals/mathematics/2023/b305106-031(2023).pdfKhuri A., Mathew T., Sinha B., Statistical Tests for Mixed Linear Models. Wiley, New York, 1998. 10.1201/9781420010435West B., Welch K., Galecki A., Linear Mixed Models: A Practical Guide Using Statistical Software. Taylor and Francis Group, LLC, 2007. 10.1098/rspa.1965.0012Nelder J. A., The analysis of randomized experiments with orthogonal block structure I, Block structure and the null analysis of variance. Proc. R. Soc. Lond. Ser. A Math., Phys. Eng. Sci., Vol.283, 1965a, pp.147–162. 10.1098/rspa.1965.0013Nelder J. A., The analysis of randomized experiments with orthogonal block structure II. Treatment structure and the general analysis of variance. Proc. R. Soc. Lond. Ser. A Math., Phys. Eng. Sci., Vol. 283, 1965b, pp.163–178. 10.12697/acutm.2008.12.01Fonseca M., Mexia J. T., Zmyślony R., Inference in normal models with commutative orthogonal block structure. Acta et Commentationes Universitatis Tartuensis de Mathematica, Vol.12, 2008, pp.3–16. 10.1016/j.laa.2006.03.045Fonseca M., Mexia J. T., Zmyślony R., Binary operations on Jordan algebras and orthogonal normal models. Linear Algebra Appl., Vol.117(1), 2006, pp.75–86. Mexia J. T., Vaquinhas R., Fonseca M., Zmyślony R., COBS: segregation, matching, crossing and nesting. in: Latest Trends and Applied Mathematics, Simulation, Modelling, 4th International Conference on Applied Mathematics, Simulation, Modelling, ASM’10. 2010, pp. 249–255. 10.1016/j.laa.2016.12.019Santos C., Nunes C., Dias C., Mexia J. T., Joining models with commutative orthogonal block structure. Linear Algebra and its Applications, Vol.517, 2017, pp.235 – 245. 10.1007/978-1-4615-7397-5_27Zmyślony R., A characterization of best linear unbiased estimators in the general linear model. Lecture Notes in Statistics, Vol.2, 1978, pp.365–373. Jordan P., Von Neumann J., Wigner E., On an algebraic generalization of the quantum mechanical formulation. Annals of Math., Vol.35 (1), 1934. 10.1080/02664763.2020.1765322Santos C., Nunes C., Dias C., Mexia J. T., Models with commutative orthogonal block structure: a general condition for commutativity. Journal of Applied Statistics, Vol. 47 (13-15), 2020, pp.2421 – 2430. 10.1214/aoms/1177696817Seely J., Linear spaces and unbiased estimation. Ann. Math. Stat., Vol.41, 1970, pp.1725 – 1734. 10.1007/s00184-014-0506-8Carvalho F., Mexia J. T., Santos C., Nunes C., Inference for types and structured families of commutative orthogonal block structures. Metrika, Vol.78, 2015, pp.337– 372. Schott J. R., Matrix Analysis for Statistics. New York: Jonh Wiley & Sons, 1997. 10.7151/dmps.1129Santos C, Ramos P, Mexia J T, Algebraic structure of step nesting designs. Discussiones Mathematicae Probability and Statistics, Vol. 30, 2010, pp.221–235. 10.1080/15598608.2009.10411942Carvalho F., Mexia J. T., Oliveira M., Estimation in Models with Commutative Orthogonal Block Structure. J. Stat. Theory Practice, Vol.3(2), 2009, pp.525–535. 10.1063/1.3497922Carvalho F., Mexia J. T., Covas R, Structured Families of Models with Commutative Orthogonal Block Structures. AIP Conf. Proc., Vol.1281, 2010, pp.1256–1259. 10.13001/1081-3810.1601Carvalho F., Mexia J. T., Santos C., Commutative orthogonal block structure and error orthogonal models. Electronic Journal of Linear Algebra, Vol25, 2013, pp.119–128. Santos C, Nunes C, Mexia J. T., OBS, COBS and mixed models associated to commutative Jordan Algebra. Bulletin of the ISI, LXII, Proceedings of 56th session of the International Statistical Institute, Lisbon, 2008, pp. 3271–3274. 10.7151/dmps.1099Carvalho F., Mexia J. T., Oliveira M. M., Canonic inference and commutative orthogonal block structure. Discussiones Mathematicae Probability and Statistics, Vol.28(2), 2008, pp. 171–181. 10.1007/978-3-030-17519-1_7Ferreira, S.S., Ferreira, D., Nunes, C., Carvalho, F., Mexia, J.T. (2019). Orthogonal Block Structure and Uniformly Best Linear Unbiased Estimators. In: Ahmed, S., Carvalho, F., Puntanen, S. (eds) Matrices, Statistics and Big Data. IWMS 2016. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-17519-1_7 pp.89-98. Mexia J. T., Nunes C., Santos C., Structured families of normal models with COBS. 17th International Workshop in Matrices and Statistics, Tomar (Portugal). Conference paper. 2008. 10.1080/09720502.2008.10700581Nunes C., Santos C., Mexia J. T., Relevant statistics for models with commutative orthogonal block structure and unbiased estimator for variance components. J. Interdiscip. Math., Vol.11, 2008, pp.553 – 564. Santos C., Nunes C., Dias C., Mexia J. T., Operations with iso-structured models with commutative orthogonal block structure: an introductory approach. Statistical Modelling on Risk Analysis: Selected contributions from ICRA9, Perugia, Italy, May 25-27, 2022. (C. P. Kitsos, T. A. Oliveira, F. Pierri, M. Restaino, eds.) Springer Proceedings in Mathematics & Statistics. eBook ISBN: 978- 3-031-39864-3 Due: 21 October 2023.