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This article deals with testing simultaneous hypotheses about the mean structure and the covariance structure in models with blocked compound symmetric (BCS) covariance structure. Considered models are used for double multivariate data, which means that m-variate vector of observation is measured repeatedly over u levels of some factor on each of n individual. Additionally, the assumption of multivariate normality for this type of data is made. We use framework of ratio of positive and negative parts of best unbiased estimators to obtain simultaneous F test. The test statistic is constructed as a ratio of test statistics for testing single hypotheses about the mean vector and the covariance matrix. In simulation study power of obtained test is compared with powers of three other F tests—two for testing single hypotheses and one for testing simultaneous hypotheses, whose test statistic is convex combination of test statistics of these two single F tests. The problem of simultaneous testing of the mean vector and covariance matrix was also consider in paper (Hyodo and Nishiyama, Commun Stat Theory Methods, 10.1080/03610926.2019.1639751, 2019).

In this article, authors consider simultaneous testing hypotheses about structure of mean vector and covariance matrix in models with some special covariance structure, so-called blocked co...

This article deals with the estimation and hypotheses testing problems for two-level and three-level multivariate data. The coordinate-free 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 for both two-level and three-level multivariate data. The estimators are shown to be unbiased, sufficient, complete, and consistent. A competitor for the likelihood ratio test on covariance components under linear constraints and the mean vectors are proposed, based on the F distribution. Simulation studies are conducted to see the power of the proposed tests and the proposed methods are implemented with two medical datasets.

The article addresses the best unbiased estimators of the block compound symmetric covariance structure for m −variate observations with equal mean vector over u sites under the assumption of multivariate quasi-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. Without assumption of normality but quasi-normality, meaning that up to fourth moments are the same as in the normal case, the estimators are best linear and best quadratic for mean and covariance parameters, respectively. Finally, strong consistency is proven. The properties of the estimators in the proposed model are compared against a similar model available in the literature. An application of the proposed approach to a clinical study data is presented.

In this article authors derive test for structure of mean vector in model with block compound symmetric covariance structure for two-level multivariate observations. One possible structure is so called structured mean vector when its components remain constant over sites or over time points, so that mean vector is of the form $\boldsymbol{1}_{u}\otimes\boldsymbol{\mu}$ with $\boldsymbol{\mu}=(\mu_1,\mu_2,\ldots,\mu_m)'\in\mathbb{R}^m$. This hypothesis is tested against alternative of unstructured mean vector, which can change over sites or over time points.

The article addresses the best unbiased estimators of doubly exchangeable covariance structure, an extension of block exchangeable covariance structure, for three-level multivariate data. Under multivariate normality, the free-coordinate approach is used to obtain linear and quadratic estimates for the model parameters that are unbiased, sufficient, complete and consistent. Data from a clinical trial study is analyzed to illustrate the application of the obtained results.

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.

The equivalence of ordinary least squares estimators (OLSE) and Gauss–Markov estimators for models with variance–covariance matrix $$\sigma ^2{\mathbf M}$$ is extended to derive a necessary and sufficient balance condition for mixed models with mean vector $${\varvec{\mu }} = {{\mathbf X} {\varvec{\beta }}}$$ , with $${\mathbf {X}}$$ an incidence matrix, having OLSE for $$\varvec{\beta }$$ that are best linear unbiased estimator whatever the variance components. This approach leads to least squares like estimators for variance components. To illustrate the range of applications for the balance condition, interesting special models are considered.

Besides the basic model, Kronecker products of rotated models are used to isolate the variance components as parameters of a linear model. A characterization of BLUE given by Zmyślony (1980) is applied to the different models. Generalized least squares are used to complete the estimation.

Commutative Jordan algebras play a central part in orthogonal models. The generations of these algebras is studied and applied in deriving lattices of such algebras. These lattices constitute the natural framework for deriving new orthogonal models through factor aggregation and disaggregation.

Given an orthogonal model $${{\bf \lambda}}=\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i$$ an L model $${{\bf y}}={\bf L}\left(\sum_{i=1}^m{{{\bf X}}_i}{\boldsymbol{\alpha}}_i\right)+{\bf e}$$ is obtained, and the only restriction is the linear independency of the column vectors of matrix L. Special cases of the L models correspond to blockwise diagonal matrices L = D(L1, . . . , Lc). In multiple regression designs this matrix will be of the form $${\bf L}={\bf D}(\check{{\bf X}}_1,\ldots,\check{{\bf X}}_{c})$$ with \({\check{{\bf X}}_j, j=1,\ldots,c}\) the model matrices of the individual regressions, while the original model will have fixed effects. In this way, we overcome the usual restriction of requiring all regressions to have the same model matrix.

Linear mixed normal models are studied in this paper. Using commutative Jordan algebras, the algebraic properties of these models are studied, as well as optimal estimators, hypothesis tests and confidence regions for fixed and random effects. Model crossing andnesting is then presented and analyzed.

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.

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.

Jordan algebras are used to present normal orthogonal models in a canonical form. Binary operations are defined on these algebras, which enable us to build complex models based on simpler ones. Properties of these operations and their relation with balanced models are studied.The canonical model formulation is interesting because it leads to complete sufficient statistics. These statistics are then used to obtain estimators in order to test hypothesis on the model parameters.

The problem of estimating the cross-product of two mean vectors in three-dimensional Euclidian space is considered. Two ‘natural’ estimators are developed, both of which turn out to be biased. A third, unbiased estimator, resulting from a jackknife procedure, is also investigated. It is shown that, under normality, the latter is best among all the unbiased estimators of this quantity.

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.

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.

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.

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.

In the paper the general linear regression model E(y)=Xβ, Cov(y)=Σki=1 σ2iVi; Vi≥0, i=1,2,…,k−1, Vk=I, is considered. For this model explicit formulae for the Bayes invariant quadratic unbiased estimators and the Bayes invariant quadratic estimators are given. These estimators are expressed in terms of a base of a Jordan algebra generated by MV1M, MV2M,…,MVk−1M, M, where M is the orthogonal projector on the null space of X′. Two-way classification random models corresponding to orthogonal and partially balanced block designs are considered as examples.

In the paper a usual block design with treatment effects fixed and block effects random is considered. Comparison of experiments is based on the asymptotic covariance matrix of a robust estimator both for shift and scale parameters given by Fisher consistent and Frechet differentiable functional (recently proposed by Bednarski and Zontek, 1994). In the class of equiblock-sized designs A- and D-optimality are discussed.

In the paper, a new approach to construction test for independence of 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 for some †-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 coecient is given. The results follow from the unified theory of testing hypotheses both for fixed eects and variance components presented in papers [6] and [7].

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).

Two main classes of estimators for parametric functions in linear models are considered, the class Γ 1 of quadratic estimators and the class Γ 2 of linear plus quadratic estimators. Let G i, i=1,2, be the class of functions for which there exist an unbiased estimator in Γ i Necessary and sufficient conditions for a given estimator in Γ i, i =1,2 to be a minimum variance estimator for its expected value are presented. Moreover, necessary and sufficient conditions for each g∈i, i =1,2,1, 2, to have an uniformly minimum variance estimator in Γ i are given. Similar problems are also discussed for 3 defined subclasses of Γ 1 and for 4 subclasses of Γ 2. Finally necessary and sufficient conditions are given for the simultaneous existence of a best linear estimator and a best quadratic estimator.

Linear models with two variance components are considered and necessary and sufficient conditions for the existence of admissible nonnegative definite quadratic estimators, biased and unbiased, are given. An unbalanced one-way random model is studied for demonstration.

In the paper we consider a class of Gaussian Markov processes y(t). For e fixed t ∈[O,T],the random vector y(t) may be interpreted as a vector of observations in a classical linear model. We give a minimal sufficient statistic and obtain a simple characterization of its completeness. Using the terminology of stochastic integrals we give explicit formulas for estimators of regression coefficients and variance components. Moreover, we prove that they are the best unbiased estimators. Such problems are considered in detection, modulation, communication and control. We use here the same notation and terminology as in [7].

A characterization of best linear unbiased estimators is given in the case of the general linear model. In addition necessary and sufficient conditions are derived for a given estimable function to have a best linear unbiased estimator. In particular models for which each estimable function has a best linear unbiased estimator are characterized. The conditions stated are given in a computational atractive form. The problems are discussed from the coordinate-free point of view. This is important for the results can be easily adopted in the case of estimation of variance components. The problem of estimation of either treatment or block effects in a mixed model serves as an example which illustrates the applicability of the results.

Let {Ω, F, Ρ} be a complete probability space and let for t Є[0,∞) $$y\left( t \right)=X\left( t \right)\beta +z\left( t \right),y\left( 0 \right)=0$$ (1) be an n-dimensional continuous stochastic process. X(t) stands for a matrix of known functions, β for a p-dimensional vector of unknown parameters and z(t) for an n-dimensional Gaussian Markov process with expectation zero and covariance matrix R(t,s).Let Ro(t)=R(t,t).