Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure and structured mean vector

Arkadiusz Kozioł

Discussiones Mathematicae Probability and Statistics (2016)

  • Volume: 36, Issue: 1-2, page 93-113
  • ISSN: 1509-9423

Abstract

top
In this article author obtain the best unbiased estimators of doubly exchangeable covariance structure. For this purpose the coordinate free-coordinate approach is used. Considered covariance structure consist of three unstructured covariance matrices for three-level m - variate observations with equal mean vector over v points in time and u sites under the assumption of multivariate normality. To prove, that the estimators are best unbiased, complete statistics are used. Additionally, strong consistency is proven. Under the proposed model the variances of the estimators of covariance components are compared with the ones in the model in [11].

How to cite

top

Arkadiusz Kozioł. "Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure and structured mean vector." Discussiones Mathematicae Probability and Statistics 36.1-2 (2016): 93-113. <http://eudml.org/doc/286961>.

@article{ArkadiuszKozioł2016,
abstract = {In this article author obtain the best unbiased estimators of doubly exchangeable covariance structure. For this purpose the coordinate free-coordinate approach is used. Considered covariance structure consist of three unstructured covariance matrices for three-level $m-$variate observations with equal mean vector over v points in time and u sites under the assumption of multivariate normality. To prove, that the estimators are best unbiased, complete statistics are used. Additionally, strong consistency is proven. Under the proposed model the variances of the estimators of covariance components are compared with the ones in the model in [11].},
author = {Arkadiusz Kozioł},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {best unbiased estimator; doubly exchangeable covariance structure; three-level multivariate data; coordinate free approach; structured mean vector},
language = {eng},
number = {1-2},
pages = {93-113},
title = {Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure and structured mean vector},
url = {http://eudml.org/doc/286961},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Arkadiusz Kozioł
TI - Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure and structured mean vector
JO - Discussiones Mathematicae Probability and Statistics
PY - 2016
VL - 36
IS - 1-2
SP - 93
EP - 113
AB - In this article author obtain the best unbiased estimators of doubly exchangeable covariance structure. For this purpose the coordinate free-coordinate approach is used. Considered covariance structure consist of three unstructured covariance matrices for three-level $m-$variate observations with equal mean vector over v points in time and u sites under the assumption of multivariate normality. To prove, that the estimators are best unbiased, complete statistics are used. Additionally, strong consistency is proven. Under the proposed model the variances of the estimators of covariance components are compared with the ones in the model in [11].
LA - eng
KW - best unbiased estimator; doubly exchangeable covariance structure; three-level multivariate data; coordinate free approach; structured mean vector
UR - http://eudml.org/doc/286961
ER -

References

top
  1. [1] M. Fonseca, J.T. Mexia and R. Zmyślony, Least squares and generalized least squares in models with orthogonal block structure, J. Statistical Planning and Inference 140 (5) (2010), 1346-1352. Zbl1181.62083
  2. [2] H. Drygas, The Coordinate-Free Approach to Gauss-Markov Estimation (Berlin, Heidelberg, Springer, 1970). Zbl0215.26504
  3. [3] S. Gnot, W. Klonecki and R. Zmyślony, Uniformly minimum variance unbiased estimation in various classes of estimators, Statistics 8 (2) (1977), 199-210. Zbl0386.62050
  4. [4] S. Gnot, W. Klonecki and R. Zmyślony, Best unbiased estimation: a coordinate free-approach, Probab. and Statis. 1 (1) (1980), 1-13. Zbl0526.62061
  5. [5] P. Jordan, J. von Neumann and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1) (1934), 29-64. Zbl60.0902.02
  6. [6] W. Kruskal, When are Gauss-Markov and Least Squares Estimators Identical, A Coordinate-Free Approach, Ann. Math. Stat. 39 (1) (1968), 70-75. Zbl0162.21902
  7. [7] E.L. Lehmann and G. Casella, Theory of Point Estimation (Second Edition, Springer, 1998). Zbl0916.62017
  8. [8] A. Roy and R. Leiva, Estimating and testing a structured covariance matrix for three-level multivariate data, Comm. Statist. Theory Methods 40 (11) (2011), 1945-1963. Zbl1216.62095
  9. [9] A. Roy and M. Fonseca, Linear models with doubly exchangeable distributed errors, Comm. Statist. Theory Methods 41 (2012), 2545-2569. Zbl1270.62102
  10. [10] A. Roy, R. Zmyślony, M. Fonseca and R. Leiva, Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure, J. Multivariate Analysis, 2016. Zbl1328.62343
  11. [11] A. Roy, A. Kozioł, R. Zmyślony, M. Fonseca and R. Leiva, Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure, Statistics 144 (2016), 81-90. Zbl1328.62343
  12. [12] J.F. Seely, Quadratic subspaces and completeness, Ann. Math. Statist. 42 (2) (1971), 710-721. Zbl0249.62067
  13. [13] J.F. Seely, Completeness for a family of multivariate normal distributions, Ann. Math. Statist. 43 (1972), 1644-1647. Zbl0257.62018
  14. [14] J.F. Seely, Minimal sufficient statistics and completeness for multivariate normal families, Sankhya (Statistics). Indian J. Statist. Ser. A 39 (2) (1977), 170-185. Zbl0409.62004
  15. [15] R. Zmyślony, On estimation of parameters in linear models, Appl. Math. XV (1976), 271-276. Zbl0401.62049
  16. [16] R. Zmyślony, A characterization of best linear unbiased estimators in the general linear model, Lecture Notes in Statistics 2 (1978), 365-373. 
  17. [17] R. Zmyślony, Completeness for a family of normal distributions, Math. Stat. Banach Center Publications 6 (1980), 355-357. Zbl0464.62003

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.