Roman Zmyślony

Testing Hypotheses for Variance Components in Mixed Linear Models

In the paper the problem of testing hypotheses for variance components in mixed linear models is considered. It is assumed that covariance matrices commute after using the usual invariance procedure with respect to the group of translations. The test for vanishing of single variance component is based on the locally best quadratic unbiased estimator of this component and rejects hypothesis if the ratio of positive and negative part of this estimator is sufficiently large. The power of this test with powers of other four tests for two-way classification models corresponding to block design is compared.

Jordan algebras, generating pivot variables and orthogonal normal models

Abstract Jordan algebras are used to present normal orthogonal models in a canonical form. It is shown that the usual factor based formulation of such models may, many times, be obtained imposing restrictions on the parameters of the canonical formulation, and examples are presented. The canonical model formulation is interesting since it leads to complete sufficient statistics. These statistics may be used to obtain pivot variables that induce probability measures in the parameter space. Monte Carlo generated samples, of arbitrary size, may be obtained having the induced probability measures. These samples may be screened so that the restrictions corresponding to the direct model formulations hold. Inference is presented using such samples.

Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure

The paper deals with the best unbiased estimators of the blocked compound symmetric covariance structure for m -variate observations over u sites under the assumption of multivariate normality. The free-coordinate approach is used to prove that the quadratic estimation of covariance parameters is equivalent to linear estimation with a properly defined inner product in the space of symmetric matrices. Complete statistics are then derived to prove that the estimators are best unbiased. Finally, strong consistency is proven. The proposed method is implemented with a real data set.

Tolerance intervals in a two-way nested model with mixed or random effects

We address the problem of deriving a one-sided tolerance interval in a two-way nested model with mixed or random effects. The generalized confidence interval idea is used in the derivation of our tolerance limit, and the results are obtained by suitably modifying the approach in Krishnamoorthy and Mathew [Krishnamoorthy, K. and Mathew, T., 2004, Generalized confidence limits and one-sided tolerance limits in balanced and unbalanced one-way random models. Technometrics, 46, 44–52], for the one-way random model. Our proposed tolerance limit can be estimated by Monte Carlo simulation. We have also been able to develop closed form approximations in some cases. The performance of our tolerance interval is numerically investigated and found to be satisfactory. The results are illustrated with an example.

Binary operations on Jordan algebras: An application to statistical inference in linear models

Necessary and sufficient conditions for the existence of best unbiased estimators (UMVUE) in Normal Linear Mixed Models, Y∼N(Xβ, ∑ i = 1kσi2Vi), are given. These conditions rely on the existence of Jordan algebras, in the sequence of [19], [6], [9], [7], [3] and [8]. We also take [2] in mind where lattices of Jordan algebras are presented for estimation purposes in these models and [1] where such estimation is exemplified.

On admissible estimation for parametric functions in linear models

This paper deals with the linear model Ey∈K, Cov y∈V. The question is investigated when a parametric function (a,y) is an admissible or inadmissible estimator of some parametric function (p,Ey). It is also discussed when a linear mapping C:K→K has the property that (a,cy) is an admissible estimator of ((Ey),a) for all a∈K. Finall the question is raised how inadmissible estimators (a,y) can be replaced by admissible estimators superior to (a,y).

Inference for the interclass correlation in familial data using small sample asymptotics

Inference on the parent-offspring correlation coefficient is an important problem in the analysis of familial data, and point estimates and likelihood based inference are available in the literature. In this work, corrections for the signed log-likelihood ratio test statistics are proposed, based on small sample asymptotics, in order to achieve accurate small sample performance. The corrected statistic can be used for hypothesis testing as well as for interval estimation.

Confidence Intervals for Mixed Log‐Normal Models

The interval estimation of a log‐normal mean is investigated when the log‐transformed data follows a regression model that also includes a random effect. The generalized confidence interval idea is used to derive the confidence interval. Two different methodologies are proposed for this problem.