Bayesian and Generalized Confidence Intervals on Variance Ratio and on the Variance Component in Mixed Linear Models

The paper deals with construction of exact confidence intervals for the variance component σ 1 and ratio θ of variance components σ 1 and σ in mixed linear models for the family of normal distributions Nt (0, σ 2 1 W + σ2It). 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 (σ 1 , σ) for: 1) the variance components ratio θ built by using test statistics obtained from the decomposition of a quadratic form yAy for the Bayes locally best estimator of σ 1 , Michalski and Zmyślony (1996), 2) the variance component σ 1 constructed using Bayes point estimators from BIQUE class (Best Invariant Quadratic Unbiased Estimators, see Gnot and Kleffe, 1983, and Michalski, 2003). 6 A. Michalski 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).