Bayesian and generalized confidence intervals on variance ratio and on the variance component in mixed linear models

Andrzej Michalski

Discussiones Mathematicae Probability and Statistics (2009)

  • Volume: 29, Issue: 1, page 5-29
  • ISSN: 1509-9423

Abstract

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The paper deals with construction of exact confidence intervals for the variance component σ₁² and ratio θ of variance components σ₁² and σ² in mixed linear models for the family of normal distributions t ( 0 , σ ₁ ² W + σ ² I t ) . This problem essentially depends on algebraic structure of the covariance matrix W (see Gnot and Michalski, 1994, Michalski and Zmyślony, 1996). In the paper we give two classes of bayesian interval estimators depending on a prior distribution on (σ₁², σ²) for: 1) the variance components ratio θ - built by using test statistics obtained from the decomposition of a quadratic form y’Ay for the Bayes locally best estimator of σ₁², Michalski and Zmyślony (1996), 2) the variance component σ₁² - constructed using Bayes point estimators from BIQUE class (Best Invariant Quadratic Unbiased Estimators, see Gnot and Kleffe, 1983, and Michalski, 2003). In the paper an idea of construction of confidence intervals using generalized p-values is also presented (Tsui and Weerahandi, 1989, Zhou and Mathew, 1994). Theoretical results for Bayes interval estimators and for some generalized confidence intervals by simulations studies for some experimental layouts are illustrated and compared (cf Arendacká, 2005).

How to cite

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Andrzej Michalski. "Bayesian and generalized confidence intervals on variance ratio and on the variance component in mixed linear models." Discussiones Mathematicae Probability and Statistics 29.1 (2009): 5-29. <http://eudml.org/doc/277011>.

@article{AndrzejMichalski2009,
abstract = {The paper deals with construction of exact confidence intervals for the variance component σ₁² and ratio θ of variance components σ₁² and σ² in mixed linear models for the family of normal distributions $_t(0, σ₁²W + σ²I_t)$. This problem essentially depends on algebraic structure of the covariance matrix W (see Gnot and Michalski, 1994, Michalski and Zmyślony, 1996). In the paper we give two classes of bayesian interval estimators depending on a prior distribution on (σ₁², σ²) for: 1) the variance components ratio θ - built by using test statistics obtained from the decomposition of a quadratic form y’Ay for the Bayes locally best estimator of σ₁², Michalski and Zmyślony (1996), 2) the variance component σ₁² - constructed using Bayes point estimators from BIQUE class (Best Invariant Quadratic Unbiased Estimators, see Gnot and Kleffe, 1983, and Michalski, 2003). In the paper an idea of construction of confidence intervals using generalized p-values is also presented (Tsui and Weerahandi, 1989, Zhou and Mathew, 1994). Theoretical results for Bayes interval estimators and for some generalized confidence intervals by simulations studies for some experimental layouts are illustrated and compared (cf Arendacká, 2005).},
author = {Andrzej Michalski},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {mixed linear models; variance components; hypothesis testing; confidence intervals; generalized p-values; generalized -values},
language = {eng},
number = {1},
pages = {5-29},
title = {Bayesian and generalized confidence intervals on variance ratio and on the variance component in mixed linear models},
url = {http://eudml.org/doc/277011},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Andrzej Michalski
TI - Bayesian and generalized confidence intervals on variance ratio and on the variance component in mixed linear models
JO - Discussiones Mathematicae Probability and Statistics
PY - 2009
VL - 29
IS - 1
SP - 5
EP - 29
AB - The paper deals with construction of exact confidence intervals for the variance component σ₁² and ratio θ of variance components σ₁² and σ² in mixed linear models for the family of normal distributions $_t(0, σ₁²W + σ²I_t)$. This problem essentially depends on algebraic structure of the covariance matrix W (see Gnot and Michalski, 1994, Michalski and Zmyślony, 1996). In the paper we give two classes of bayesian interval estimators depending on a prior distribution on (σ₁², σ²) for: 1) the variance components ratio θ - built by using test statistics obtained from the decomposition of a quadratic form y’Ay for the Bayes locally best estimator of σ₁², Michalski and Zmyślony (1996), 2) the variance component σ₁² - constructed using Bayes point estimators from BIQUE class (Best Invariant Quadratic Unbiased Estimators, see Gnot and Kleffe, 1983, and Michalski, 2003). In the paper an idea of construction of confidence intervals using generalized p-values is also presented (Tsui and Weerahandi, 1989, Zhou and Mathew, 1994). Theoretical results for Bayes interval estimators and for some generalized confidence intervals by simulations studies for some experimental layouts are illustrated and compared (cf Arendacká, 2005).
LA - eng
KW - mixed linear models; variance components; hypothesis testing; confidence intervals; generalized p-values; generalized -values
UR - http://eudml.org/doc/277011
ER -

References

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  15. [15] K.W. Tsui and S. Weerahandi, Generalized P-values in significance testing of hypotheses in the presence of nuisance parameters, J. Amer. Statist. Assoc. 84 (1989), 602-607. 
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