ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

Contributed talks on Block Orthogonalization

Mon 14:00–14:30, Room AV 00.17, Chair: Jesse L. Barlow
Optimal Estimation in Models with Positive Orthogonal Block Structure
Sandra Ferreira (University of Beira Interior, Covilhã, Portugal)
Joint work with Dário Ferreira; Célia Nunes; João Tiago Mexia

The aim of this paper is to obtain estimators for estimable vectors that are UMVUE [UBLUE] when normality is assumed [in general] under conditions that do not involve the orthogonal projection matrix $T$ on the space $\Omega$ spanned by the mean vector. Namely we consider models whose variance-covariance matrices are all the positive definite linear combinations $V(\gamma)= \sum_{j=1}^{m} \gamma_{j} K_{j},$ where the $K_{1},...,K_{m}$ are pairwise orthogonal orthogonal projection matrices, POOPM, that add up to $I_{n}.$ So that $\gamma \in \mathbb{R}^{m}_{>},$ with $\mathbb{R}^{m}_{>}$ $[\mathbb{R}^{m}_{\geq}]$ the family of vectors with positive [non negative] components of subspace $\mathbb{R}^{m}.$ If we only required $V(\gamma)$ to be semi-positive definite we would have $\gamma \in \mathbb{R}^{m}_{\geq},$ and the model, would have orthogonal block structure, OBS. We may say that the models we are considering have positive OBS, POBS.

Application to a real data set is also presented.

1. Ferreira, S. S. ; Ferreira, D. ; Fernandes, C. and Mexia, J.T. . Orthogonal Mixed Models and Perfect Families of Symmetric Matrices. Book of Abstracts. 56-th Session of the International Statistical Institute, ISI, Lisboa, Pg. 291, 22 a 29 de Agosto (2007).
2. Fonseca, M.; Mexia, J. T. and Zmyślony, R. . Estimating and Testing of Variance Components: An Application to a Grapevine Experiment. Biometrical Letters 40, 1, 1–7 (2003).
3. Fonseca, M., Mexia, J.T. and Zmyślony, R. . Binary operations on Jordan algebras and orthogonal normal models, Linear Algebra Appl. 417, 75–86 (2006).
4. Michalski, A., Zmyślony, R. . Testing Hypotheses for Variance Components in Mixed Linear Models Statistics 27(3-4), 297–310 (1996).
5. Michalsky, A., Zmyślony, R.. Testing hypotheses for linear functions of parameters in mixed linear models, Tatra Mountain Mathematical Publications, 17, 103–110 (1999).
6. Nelder, J.A. . The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proceedings of the Royal Society (London), Series A 273, 147–162 (1965a).
7. Nelder, J.A.. The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proceedings of the Royal Society (London), Series A 273, 163–178 (1965b).
8. Rao, C. R., Kleffé. J. . Estimation of Variance Components and Its Applications. Amsterdam, North Holland (1988).
9. Silvey, S.D. . Statistical Inference. Reprinted. Chapman & Hall (1975).
10. Seely, J. . Quadratic subspaces and completeness. Ann. Statist. 42, 2, 710–721 (1971).
11. Schott, J. R. . Matrix Analysis for Statistics. Jonh Wiley & Sons, New York (1997).
12. VanLeeuwen, D.M., Seely, J.F. and Birkes, D.S. . Sufficient conditions for orthogonal designs in mixed linear models. J. Stat. Plan. Inference 73, 373–389 (1998).
13. Zmyślony, R. . A characterization of best linear unbiased estimators in the general linear model, Mathematical Statistics and Probability Theory, Proc. Sixth Internat. Conf., Wisla, Lecture Notes in Statist., Springer, New York-Berlin 2, 365–373 (1978).
14. Zyskind, G. . On canonical forms, non-negative covariance matrices and best simple linear least squares estimators in linear models. Ann. Math. Stat. 38, 1092–1109 (1967).