Orthogonal models: Algebraic structure and explicit estimators for estimable vectors

Artur Pereira; Miguel Fonseca; João Tiago Mexia

Discussiones Mathematicae Probability and Statistics (2015)

  • Volume: 35, Issue: 1-2, page 29-44
  • ISSN: 1509-9423

Abstract

top
We study the algebraic structure of orthogonal models thus of mixed models whose variance covariance matrices are all positive semi definite, linear combinations of known pairwise orthogonal projection matrices, POOPM, and whose least square estimators, LSE, of estimable vectors are best linear unbiased estimator, BLUE, whatever the variance components, so they are uniformly BLUE, UBLUE. From the results of the algebraic structure we will get explicit expression for the LSE of these models.

How to cite

top

Artur Pereira, Miguel Fonseca, and João Tiago Mexia. "Orthogonal models: Algebraic structure and explicit estimators for estimable vectors." Discussiones Mathematicae Probability and Statistics 35.1-2 (2015): 29-44. <http://eudml.org/doc/276480>.

@article{ArturPereira2015,
abstract = {We study the algebraic structure of orthogonal models thus of mixed models whose variance covariance matrices are all positive semi definite, linear combinations of known pairwise orthogonal projection matrices, POOPM, and whose least square estimators, LSE, of estimable vectors are best linear unbiased estimator, BLUE, whatever the variance components, so they are uniformly BLUE, UBLUE. From the results of the algebraic structure we will get explicit expression for the LSE of these models.},
author = {Artur Pereira, Miguel Fonseca, João Tiago Mexia},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {linear models; mixed models; inference; orthogonal models; UBLUE},
language = {eng},
number = {1-2},
pages = {29-44},
title = {Orthogonal models: Algebraic structure and explicit estimators for estimable vectors},
url = {http://eudml.org/doc/276480},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Artur Pereira
AU - Miguel Fonseca
AU - João Tiago Mexia
TI - Orthogonal models: Algebraic structure and explicit estimators for estimable vectors
JO - Discussiones Mathematicae Probability and Statistics
PY - 2015
VL - 35
IS - 1-2
SP - 29
EP - 44
AB - We study the algebraic structure of orthogonal models thus of mixed models whose variance covariance matrices are all positive semi definite, linear combinations of known pairwise orthogonal projection matrices, POOPM, and whose least square estimators, LSE, of estimable vectors are best linear unbiased estimator, BLUE, whatever the variance components, so they are uniformly BLUE, UBLUE. From the results of the algebraic structure we will get explicit expression for the LSE of these models.
LA - eng
KW - linear models; mixed models; inference; orthogonal models; UBLUE
UR - http://eudml.org/doc/276480
ER -

References

top
  1. [1] A. Areia and F. Carvalho, Perfect families: an appliacation to orthogonal and error orthogonal models, AIP Conf. Proc. Numerical Analysis and Applied Mathematics: International Conferences of Numerical Analysis and Applied Mathematics 1558 (2013), 841-846. 
  2. [2] T. Calinski and S. Kageyama, Block Designs: A Randomization Approach. Vol. I: Analysis , Lecture Notes in Statistics, Springer, 2000, 150. 
  3. [3] T. Calinski and S. Kageyama, Block Designs: A Randomization Approach. Vol. II: Design, Lecture Notes in Statistics, Springer, 2003, 170. 
  4. [4] F. Carvalho, J.T. Mexia and M.M. Oliveira, Canonic inference and commutative orthogonal block structure, Discuss. Math. Probab. and Stat. 28 (2) (2008), 171-181. Zbl1208.62093
  5. [5] F. Carvalho, J.T. Mexia and C. Santos, Commutative orthogonal block structure and error orthogonal models, Electronic Journal of Linear Algebra 25 (2013), 119-128. Zbl1283.15087
  6. [6] S.S. Ferreira, D. Ferreira, C. Fernandesa and J.T. Mexia, Orthogonal models and perfect families of symmetric matrices, Bulletin of the International Statistical Institute. Proc. ISI 2007, Lisboa 22-28 August (2007), 3252-3254. 
  7. [7] M. Fonseca, J.T. Mexia and R. Zmyślony, Inference in normal models with commutative orthogonal block structure, Acta et Commentationes Universitatis Tartunesis de Mathematica (2008), 3-16. Zbl1166.62313
  8. [8] A. Houtman and T.P. Speed, Balance in designed experiments with orthogonal block structure, The annals of Statistics 11 (4) (1983), 1069-1085. Zbl0566.62065
  9. [9] E.L. Lehmann and G. Casela, Theory of Point estimation, 2nd ed. (Springer Tests Statistics, New York, Springer, 1998). 
  10. [10] S. Mejza, On some aspects of general balance in designed experiments, Statistica 52 (1992), 263-278. Zbl0770.62060
  11. [11] J.T. Mexia, R. Vaquinhas, M. Fonseca and R. Zmyślony, COBS: segregation, matching, crossing and nesting, Latest Trends on Applied Mathematics, Simulation, Modelling (2010), 249-255. 
  12. [12] J.A. Nelder, The analysis of randomized experiments with orthogonal block structure and the null analysis of variance, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 283 (1393), 147-162. Zbl0124.10703
  13. [13] J.A. Nelder, The analysis of randomized experiments with orthogonal block structure II. Treatment structure and the general analysis of variance, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 283 (1393), 163-178. Zbl0124.10703
  14. [14] J.R. Schott, Matrix Analysis for Statistics (Wiley Series in Probability and Statistics, 1997). Zbl0872.15002
  15. [15] J. Seely, Quadratic subspaces and completeness, The Annals of Mathematical Statistics 42 (1971), 710-721. Zbl0249.62067
  16. [16] D.M. Vanleeuwen, D.S. Birks, J.F. Seely, J. Mills, J.A. Greenwood and C.W. Jones, Sufficient conditions for orthogonal designs in mixed linear models, Journal of Statistics Planning and Inference 73 (1998), 373-389. 
  17. [17] R. Zmyślony, A characterization of the best linear unbiased estimators in the general linear model, Lecture Notes Statistics 2 (1978), 365-373. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.