Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 28, 2014

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

  • Aníbal Areia , Francisco Carvalho EMAIL logo and João T. Mexia
From the journal Open Mathematics


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.


[1] Carvalho, Francisco; Mexia, João T.; Oliveira, M. Manuela, Estimation in Models with Commutative Orthogonal Block Structure, J. Stat. Theory Pract., 2009, 3 (2), 525-53510.1080/15598608.2009.10411942Search in Google Scholar

[2] Drygas, H., Sufficiency and Completeness in the General Gauss-Markov Model, Sankhy¯ a, 1983, 45 (1), 88-89Search in Google Scholar

[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, 3252-3254Search in Google Scholar

[4] Fonseca, M; Mexia, João T.; Zmy´slony, R., Binary operations on Jordan algebras and orthogonal normal models, Linear Algebra Appl., 2006, 417, 75-8610.1016/j.laa.2006.03.045Search in Google Scholar

[5] Jordan, P.; von Neumann, J. and Wigner, E., On the algebraic generalization of the quantum mechanical formalism, Ann. of Math., 1934, 36, 26-64Search in Google Scholar

[6] Lehmann, E.L. and Casella, G., Theory of Point Estimation, 2nd ed., Springer, 1998Search in Google Scholar

[7] Schott, James R., Matrix Analysis for Statistics, Wiley Series in Probability and Statistics, 1997Search in Google Scholar

[8] Seely, J., Linear spaces and unbiased estimators, Ann. Math. Stat., 1970a, 41, 1735-174510.1214/aoms/1177696818Search in Google Scholar

[9] Seely, J., Linear spaces and unbiased estimators. Application to a mixed linear model, Ann. Math. Stat., 1970b, 41, 1735-174510.1214/aoms/1177696818Search in Google Scholar

[10] Seely, J., Quadratic subspaces and completeness, Ann. Math. Stat., 1971a, 42, 710-72110.1214/aoms/1177693420Search in Google Scholar

[11] Seely, J., Zyskind, Linear spaces and minimum variance estimators, Ann. Math. Stat., 1971b, 42, 691-70310.1214/aoms/1177693418Search in Google Scholar

[12] Seely, J., Minimal sufficient statistics and completeness for multivariate normal families, Sankhy¯ a, 1977, 39 (2), 170-185Search in Google Scholar

[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, 373-38910.1016/S0378-3758(98)00071-8Search in Google Scholar

[14] VanLeeuwen, Dawn M.; Birkes, David S.; Seely, Justus F., Balance and Orthogonality in Designs for Mixed Classification Models, Ann. Statist., 1999, 27 (6), 1927-194710.1214/aos/1017939245Search in Google Scholar

[15] Zmy´slony, R; Drygas, H., Jordan Algebras and Bayesian Quadratic Estimation of Variance Components, Linear Algebra Appl., 1992, 168, 259-27510.1016/0024-3795(92)90297-NSearch in Google Scholar

Received: 2013-12-9
Accepted: 2014-4-3
Published Online: 2014-10-28
Published in Print: 2015-1-1

© 2015 Aníbal Areia et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 27.2.2024 from
Scroll to top button