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Selective generalized F tests

100%

Nunes C., Mexia J.

Discussiones Mathematicae Probability and Statistics

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2004

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tom 24

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nr 2

281-288

EN

Generalized F tests were introduced by Michalski and Zmyślony (1996) for variance components and later (1999) for linear functions of parameters in mixed linear models. We now use generalized polar coordinates to obtain, for the second case, tests that are more powerful for selected families of alternatives.

2

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

88%

Michalski A.

Discussiones Mathematicae Probability and Statistics

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2009

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tom 29

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nr 1

5-29

EN

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).

3

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

88%

Gnot S., Michalski A., Urbańska-Motyka A.

Discussiones Mathematicae Probability and Statistics

|

2004

|

tom 24

|

nr 1

109-126

EN

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.

4

Tests of independence of normal random variables with known and unknown variance ratio

75%

Gąsiorek E., Michalski A., Zmyślony R.

Discussiones Mathematicae Probability and Statistics

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2000

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tom 20

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nr 2

233-247

EN

In the paper, a new approach to construction test for independenceof two-dimensional normally distributed random vectors is given under the assumption that the ratio of the variances is known. This test is uniformly better than the t-Student test. A comparison of the power of these two tests is given. A behaviour of this test forsome ε-contamination of the original model is also shown. In the general case when the variance ratio is unknown, an adaptive test is presented. The equivalence between this test and the classical t-test for independence of normal variables is shown. Moreover, the confidence interval for correlation coefficient is given. The results follow from the unified theory of testing hypotheses both for fixed effects and variance components presented in papers [6] and [7].

Ograniczanie wyników

4 Discussiones Mathematicae Probability and Statistics

3 Michalski A.

1 Gnot S.

1 Gąsiorek E.

1 Mexia J.

1 Nunes C.

1 Urbańska-Motyka A.

1 Zmyślony R.

1 2009

2 2004

1 2000

Selective generalized F tests

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

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

Tests of independence of normal random variables with known and unknown variance ratio