On some properties of ML and REML estimators in mixed normal models with two variance components

Stanisław Gnot; Andrzej Michalski; Agnieszka Urbańska-Motyka

Discussiones Mathematicae Probability and Statistics (2004)

  • Volume: 24, Issue: 1, page 109-126
  • ISSN: 1509-9423

Abstract

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In the paper, the problem of estimation of variance components σ₁² and σ₂² by using the ML-method and REML-method in a normal mixed linear model 𝒩 {Y,E(Y) = Xβ, Cov(Y) = σ₁²V + σ₂²Iₙ} is considered. This paper deal with properties of estimators of variance components, particularly when an explicit form of these estimators is unknown. The conditions when the ML and REML estimators can be expressed in explicit forms are given, too. The simulation study for one-way classification unbalanced random model together with a new proposition of approximation of expectation and variances of ML and REML estimators are shown. Numerical calculations with reference to the generalized Fisher's information are also given.

How to cite

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Stanisław Gnot, Andrzej Michalski, and Agnieszka Urbańska-Motyka. "On some properties of ML and REML estimators in mixed normal models with two variance components." Discussiones Mathematicae Probability and Statistics 24.1 (2004): 109-126. <http://eudml.org/doc/287723>.

@article{StanisławGnot2004,
abstract = {In the paper, the problem of estimation of variance components σ₁² and σ₂² by using the ML-method and REML-method in a normal mixed linear model 𝒩 \{Y,E(Y) = Xβ, Cov(Y) = σ₁²V + σ₂²Iₙ\} is considered. This paper deal with properties of estimators of variance components, particularly when an explicit form of these estimators is unknown. The conditions when the ML and REML estimators can be expressed in explicit forms are given, too. The simulation study for one-way classification unbalanced random model together with a new proposition of approximation of expectation and variances of ML and REML estimators are shown. Numerical calculations with reference to the generalized Fisher's information are also given.},
author = {Stanisław Gnot, Andrzej Michalski, Agnieszka Urbańska-Motyka},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {mixed linear models; likelihood-based inference; ML- and REML- estimation; variance components; Fisher's information; Fisher information; restricted maximum likelihood; simulations},
language = {eng},
number = {1},
pages = {109-126},
title = {On some properties of ML and REML estimators in mixed normal models with two variance components},
url = {http://eudml.org/doc/287723},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Stanisław Gnot
AU - Andrzej Michalski
AU - Agnieszka Urbańska-Motyka
TI - On some properties of ML and REML estimators in mixed normal models with two variance components
JO - Discussiones Mathematicae Probability and Statistics
PY - 2004
VL - 24
IS - 1
SP - 109
EP - 126
AB - In the paper, the problem of estimation of variance components σ₁² and σ₂² by using the ML-method and REML-method in a normal mixed linear model 𝒩 {Y,E(Y) = Xβ, Cov(Y) = σ₁²V + σ₂²Iₙ} is considered. This paper deal with properties of estimators of variance components, particularly when an explicit form of these estimators is unknown. The conditions when the ML and REML estimators can be expressed in explicit forms are given, too. The simulation study for one-way classification unbalanced random model together with a new proposition of approximation of expectation and variances of ML and REML estimators are shown. Numerical calculations with reference to the generalized Fisher's information are also given.
LA - eng
KW - mixed linear models; likelihood-based inference; ML- and REML- estimation; variance components; Fisher's information; Fisher information; restricted maximum likelihood; simulations
UR - http://eudml.org/doc/287723
ER -

References

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  1. [1] D. Birkes and S.S. Wulff, Existence of maximum likelihood estimates in normal variance-components models, J. Statist. Plann. Inference 113 (2003), 35-47. Zbl1031.62042
  2. [2] T. Caliński and S. Kageyama, Block Designs: A Randomization Approach, Vol. I: Analysis, Lecture Notes in Statistics, Springer 2000. 
  3. [3] M.Y. Elbassouni, On the existency of explicit restricted maximim likelihood estimators in multivariate normal models, Sankhyā Series B 45 (1983), 303-305. 
  4. [4] S. Gnot, D. Stemann, G. Trenkler and A. Urbańska-Motyka, On maximum likelihood estimators in multivariate normal mixed models with two variance components, Tatra Mountains Mathematical Publications 17 (1999), 111-119. Zbl0987.62041
  5. [5] S. Gnot, D. Stemann, G. Trenkler and A. Urbańska-Motyka, Maximum likelihood estimators in mixed normal models with two variance components, Statistics 36 (4) (2002), 283-302. Zbl1003.62017
  6. [6] E.L. Lehman, Theory of Point Estimation, Wiley, New York 1983. 
  7. [7] A. Olsen, J. Seely and D. Birkes, Invariant quadratic unbiased estimation for two variance components, Ann. Statist. 4 (1976), 878-890. Zbl0344.62060
  8. [8] H.D. Patterson and R. Thompson, Recovery of intra-block information when block sizes are unequal, Biometrika 58 (1971), 545-554. Zbl0228.62046
  9. [9] H.D. Patterson and R. Thompson, Maximum likelihood estimation of components of variance, Proceedings of 8th Biometric Conference (1975), 197-207. Zbl0332.62053
  10. [10] C.R. Rao, MINQUE theory and its relation to ML and MML estimation of variance components, Sankhyā Series B 41 (1979), 138-153. Zbl0485.62066
  11. [11] C.R. Rao and J. Kleffe, Estimation of Variance Components and Applications, North Holland, Amsterdam 1988. Zbl0645.62073
  12. [12] Poduri S.R.S. Rao, Variance Components Estimation, Mixed Models, Methodologies and Applications, Chapman and Hall, London, Weinheim, New York, Tokyo, Melbourne, Madras 1997. Zbl0996.62501
  13. [13] S.R. Searle, G. Casella and Ch.E. McCulloch, Variance Components, Wiley, New York 1992. 
  14. [14] J. Seely, Minimal sufficient statistics and completeness for multivariate normal families, Sankhyā Ser.A 39 (1977), 170-185. Zbl0409.62004
  15. [15] G.W. Snedecor and W.G. Cochran, Statistical Methods (8th ed.), Ames, IA: Iowa State University Press 1989. Zbl0727.62003
  16. [16] S.E. Stern and A.H. Welsh, Likelihood inference for small variance components, The Canadian Journal of Statistics 28 (2000), 517-532. Zbl0958.62025
  17. [17] W.H. Swallow and J.F. Monahan, Monte Carlo comparison of ANOVA, MIVQUE, REML, and ML estimators of variance components, Technometrics 26 (1984), 4, 7-57. Zbl0548.62051
  18. [18] T.H. Szatrowski, Necessary and sufficient conditions for explicit solutions in the multivariate normal estimation problem for patterned means and covariances, Ann. Statist. 8 (1980), 803-810. Zbl0497.62045
  19. [19] T.H. Szatrowski and J.J. Miller, Explicit maximum likelihood estimates from balanced data in the mixed model of the analysis of variance, Ann. Statist. 8 (1980), 811-819. Zbl0465.62069

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