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2009 | 29 | 1 | 5-29
Tytuł artykułu

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

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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 $𝒩_t(0, σ₁²W + σ²I_t)$. 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).
Twórcy
  • Department of Mathematics, Wrocław University of Environmental and Life Sciences, Grunwaldzka 53, 50–357 Wrocław, Poland
Bibliografia
  • [1] B. Arendacká, Generalized confidence intervals on the variance component in mixed linear models with two variance components, Statistics 39 (4) (2005), 275-286.
  • [2] R.K. Burdick and F.A. Graybill, Confidence intervals on variance components, Marcel Dekker, Inc., New York, Basel, Hong Kong 1992.
  • [3] R.B. Davies, The distribution of a linear combination of χ² random variables, Applied Statistics 29 (1980), 323-333.
  • [4] S. Gnot, Bayes estimation in linear models: A cordinate - free approach, J. Mulivariate Anal. 13 (1983), 40-51.
  • [5] S. Gnot and J. Kleffe, Quadratic estimation in mixed linear models with two variance components, J. Statist. Plann. Inference 8 (1983), 267-279.
  • [6] S. Gnot, Estimation of variance components in linear models. Theory and applications, WNT, Warszawa 1991 (in Polish).
  • [7] S. Gnot and A. Michalski, Tests based on admissible estimators in two variance components models, Statistics 25 (1994), 213-223.
  • [8] J.P. Imhof, Computing the distribution of quadratic forms in normal variables, Biometrika 48 (1961), 419-426.
  • [9] A.I. Khuri, T. Mathew and B.K. Sinha, Statistical Tests for Mixed Linear Models, Wiley & Sons, New York, Toronto 1998.
  • [10] A. Michalski, Confidence intervals on the variance ratio in two variance components models, Discussiones Mathematicae - Algebra and Stochastic Methods 15 (1995), 179-188.
  • [11] A. Michalski and R. Zmyślony, Testing hypotheses for variance components in mixed linear models, Statistics 27 (1996), 297-310.
  • [12] A. Olsen, J. Seely and D. Birkes, Invariant quadratic unbiased estimation for two variance components, Ann. Statist. 4 (1976), 878-890.
  • [13] J. Seely, Minimal sufficient statistics and completeness for multivariate normal families, Sankhyā A 39 (1977), 170-185.
  • [14] J. Seely and Y. El-Bassiouni, Applying Wald's variance components test, Ann. Statist. 11 (1983), 197-201.
  • [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.
  • [16] S. Weerahandi, Testing variance components in mixed linear models with generalized P-values, J. Amer. Statist. Assoc. 86 (1991), 151-153.
  • [17] S. Weerahandi, Generalized confidence intervals, J. Amer. Statist. Assoc. 88 (1993), 899-905.
  • [18] S. Weerahandi, Exact Statistical Methods for Data Analysis, Springer-Verlag, New York 1995.
  • [19] L. Zhou and T. Mathew, Some tests for variance components using generalized p-values, Technometrics 36 (1994), 394-402.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1104
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