Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models

Aníbal Areia; Francisco Carvalho; João T. Mexia

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 135-140, electronic only
  • ISSN: 2391-5455

Abstract

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We will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters. Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters.

How to cite

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Aníbal Areia, Francisco Carvalho, and João T. Mexia. "Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models." Open Mathematics 13.1 (2015): 135-140, electronic only. <http://eudml.org/doc/268731>.

@article{AníbalAreia2015,
abstract = {We will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters. Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters.},
author = {Aníbal Areia, Francisco Carvalho, João T. Mexia},
journal = {Open Mathematics},
keywords = {Orthogonal models; Perfect families; Commutative Jordan algebras; orthogonal models; perfect families; commutative Jordan algebras},
language = {eng},
number = {1},
pages = {135-140, electronic only},
title = {Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models},
url = {http://eudml.org/doc/268731},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Aníbal Areia
AU - Francisco Carvalho
AU - João T. Mexia
TI - Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 135
EP - 140, electronic only
AB - We will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters. Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters.
LA - eng
KW - Orthogonal models; Perfect families; Commutative Jordan algebras; orthogonal models; perfect families; commutative Jordan algebras
UR - http://eudml.org/doc/268731
ER -

References

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  1. [1] Carvalho, Francisco; Mexia, João T.; Oliveira, M. Manuela, Estimation in Models with Commutative Orthogonal Block Structure, J. Stat. Theory Pract., 2009, 3 (2), 525-535 Zbl1211.62093
  2. [2] Drygas, H., Sufficiency and Completeness in the General Gauss-Markov Model, Sankhy¯ a, 1983, 45 (1), 88-89 Zbl0535.62007
  3. [3] Ferreira, S.S.; Ferreira, D.; Fernandes, C.; Mexia, João T., Orthogonal models and perfect families of symmetric matrices, Bulletin of the ISI, Proceedings of ISI (22-28 August 2007, Lisbon, Portugal), Lisbon, 2007, 3252-3254 
  4. [4] Fonseca, M; Mexia, João T.; Zmy´slony, R., Binary operations on Jordan algebras and orthogonal normal models, Linear Algebra Appl., 2006, 417, 75-86 Zbl1113.62004
  5. [5] Jordan, P.; von Neumann, J. and Wigner, E., On the algebraic generalization of the quantum mechanical formalism, Ann. of Math., 1934, 36, 26-64 Zbl60.0902.02
  6. [6] Lehmann, E.L. and Casella, G., Theory of Point Estimation, 2nd ed., Springer, 1998 Zbl0916.62017
  7. [7] Schott, James R., Matrix Analysis for Statistics, Wiley Series in Probability and Statistics, 1997 
  8. [8] Seely, J., Linear spaces and unbiased estimators, Ann. Math. Stat., 1970a, 41, 1735-1745[Crossref] Zbl0263.62041
  9. [9] Seely, J., Linear spaces and unbiased estimators. Application to a mixed linear model, Ann. Math. Stat., 1970b, 41, 1735-1745[Crossref] Zbl0263.62041
  10. [10] Seely, J., Quadratic subspaces and completeness, Ann. Math. Stat., 1971a, 42, 710-721[Crossref] Zbl0249.62067
  11. [11] Seely, J., Zyskind, Linear spaces and minimum variance estimators, Ann. Math. Stat., 1971b, 42, 691-703[Crossref] Zbl0217.51602
  12. [12] Seely, J., Minimal sufficient statistics and completeness for multivariate normal families, Sankhy¯ a, 1977, 39 (2), 170-185 Zbl0409.62004
  13. [13] VanLeeuwen, Dawn M.; Seely, Justus F.; Birkes, David S., Sufficient conditions for orthogonal designs in mixed linear models, J. Statist. Plann. Inference, 1998, 73, 373-389 Zbl0933.62069
  14. [14] VanLeeuwen, Dawn M.; Birkes, David S.; Seely, Justus F., Balance and Orthogonality in Designs for Mixed Classification Models, Ann. Statist., 1999, 27 (6), 1927-1947 Zbl0963.62059
  15. [15] Zmy´slony, R; Drygas, H., Jordan Algebras and Bayesian Quadratic Estimation of Variance Components, Linear Algebra Appl., 1992, 168, 259-275 Zbl0760.62068

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