Asymmetrical Orthogonal Arrays With Run Size 100

Nowadays orthogonal arrays play important roles in statistics and other fields. Usual difference matrices are essential for the construction of many symmetrical or a few asymmetrical orthogonal arrays. But there are also asymmetrical orthogonal arrays which can not be obtained by the usual difference matrices. In order to construct these asymmetrical orthogonal arrays, a class of special matrices were discovered from the orthogonal decompositions of projection matrices. In this article, an interesting equivalent relationship between orthogonal arrays and the generalized difference matrices is presented. As an application, a lot of new orthogonal arrays of run size 100 have been constructed.     

A permutation procedure for testing the equality of pattern hypotheses across groups involving correlation or covariance matrices

This paper describes a permutation procedure to test for the equality of selected elements of a covariance or correlation matrix across groups. It involves either centring or standardising each variable within each group before randomly permuting observations between groups. Since the assumption of exchangeability of observations between groups does not strictly hold following such transformations, Monte Carlo simulations were used to compare expected and empirical rejection levels as a function of group size, the number of groups and distribution type (Normal, mixtures of Normals and Gamma with various values of the shape parameter). The Monte Carlo study showed that the estimated probability levels are close to those that would be obtained with an exact test except at very small sample sizes (5 or 10 observations per group). The test appears robust against non-normal data, different numbers of groups or variables per group and unequal sample sizes per group. Power was increased with increasing sample size, effect size and the number of elements in the matrix and power was decreased with increasingly unequal numbers of observations per group.     

Testing the equality of correlation matrices

We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent x ² random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative using a linear combination of independent non-central x ² random variables. We also investigate the null and the alternative distribution of two related statistics. The first one is a quadratic form in deviations from a control group with which the remaining k-1 groups are to be compared. The second test is designed for comparing adjacent groups. Several approximations for the null and the alternative distribution are considered and two illustrative examples are provided.     

On the non-existence of generalised Hadamard matrices

The determinant of a generalized Hadamard matrix over its group ring factored out by the relation Σ_gεG G = 0 is shown to have certain number theoretic properties. These are exploited to prove the non-existence of many generalised Hadamard matrices for groups whose orders are divisible by 3, 5 or 7. For example the GH(15, C₁₅), GH(15, C₃) and GH(15, C₅) do not exist. Also for certain n and G we find the set of determinants of the GH(n, G) matrices.     

A solution to the multivariate behrens-fisher problem

Some new algebra on pattern and transition matrices is used to determine the degrees of freedom and the parameter matrix, if the distribution of a linear sum of Wishart matrices is approximated by a single Wishart distribution. This approximation is then used to find a solution to the multivariate Behrens-Fisher problem similar to the Welch (1947) solution in the univariate case.     

Multivariate normal estimation: the case (n < p)

Estimation in the multivariate context when the number of observations available is less than the number of variables is a classical theoretical problem. In order to ensure estimability, one has to assume certain constraints on the parameters. A method for maximum likelihood estimation under constraints is proposed to solve this problem. Even in the extreme case where only a single multivariate observation is available, this may provide a feasible solution. It simultaneously provides a simple, straightforward methodology to allow for specific structures within and between covariance matrices of several populations. This methodology yields exact maximum likelihood estimates.     

An Improved Result Relating Quadratic Forms and Chi-Square Distributions

Results from the theory of linear models establish a particular idempotency condition as being necessary and sufficient for a quadratic form in a nonsingular normal vector to follow a chi-square distribution. We give a theorem that is somewhat stronger than the standard ones, and provide a proof that is more accessible than those usually given, in that it uses only linear algebra and calculus.     

Optimal weighing designs with a string property

We consider the usual (spring balance) weighing design set-up with the design matrix having a string property meaning thereby that in every row of it, there is exactly one run of 1's (the rest of the elements being 0's). We have investigated some interesting features of such matrices and used them in deriving various optimality results.     

Multi-sample test for high-dimensional covariance matrices

In this paper, we propose a new test statistic for testing the equality of high-dimensional covariance matrices for multiple populations. The proposed test statistic generalizes the test of the equality of two population covariance matrices proposed by Li and Chen (2012).     

A simple way to obtain the spectral decomposition of variance components models for balanced data

By means of an example it is shown how eigenvalues and eigenvectors of variance components models can be obtained straightforwardly when balanced data are available. Simple asymptotically efficient estimators of the variance components are presented.     

Supersaturated designs of projectivity P=3 or near P=3

Supersaturated designs offer a potentially useful way to investigate many factors in few experiments i.e. typical screening situations. Their design properties have mainly been evaluated based on their ability to identify and estimate main effects. Projective properties have received little attention. In this paper we show how to construct two-level supersaturated designs for 2(n−2) factors in n runs (n a multiple of four) of projectivity P=3 or near projectivity P=3 from orthogonal non-regular two-level designs. The designs obtained also have favourable properties such as low maximum absolute value of the inner product between a main effect column and a two-factor interaction column and relatively few types of different projections onto subsets consisting of three factor columns.     

Spatial circulants, with applications

The cumulants of quadratic forms associated to the so-called spatial design matrices are often needed for inference in the context of isotropic processes on uniform grids. Because the eigenvalues of the matrices involved are generally unknown, the computation of the cumulants can be very demanding if the grids are large. This paper first replaces the spatial design matrices with circular counterparts having known eigenvalues. It then studies some of the properties of the approximating matrices, and analyzes their performance in a number of applications to well-known inferential procedures.     

Maximum likelihood estimation of parameter structures for the Wishart distribution using constraints

Maximum likelihood estimation under constraints for estimation in the Wishart class of distributions, is considered. It provides a unified approach to estimation in a variety of problems concerning covariance matrices. Virtually all covariance structures can be translated to constraints on the covariances. This includes covariance matrices with given structure such as linearly patterned covariance matrices, covariance matrices with zeros, independent covariance matrices and structurally dependent covariance matrices. The methodology followed in this paper provides a useful and simple approach to directly obtain the exact maximum likelihood estimates. These maximum likelihood estimates are obtained via an estimation procedure for the exponential class using constraints.     

Ellipsoidal confidence regions for a normal covariance matrix

We obtain an asymptotic expansion of the confidence coefficient for an ellipsoidal confidence region on the elements of a normal covariance matrix. This leads to simultaneous confidence intervals on all linear functions of the elements of this matrix, which are compared with those of Roy (1954).     

On estimating the covariance matrix of robust regression M-estimates

By analyzing a special class of regression problems we point out that previously suggested estimates of the covariance matrix of regression M-cstimatos are inadequate for certain design matrices. These results confirm the conclusions drawn in several Monte Cailo studies.     

Limiting Spectral Distribution for Large Sample Covariance Matrices with m-Dependent Elements

This article establishes the limiting spectral distribution of large sample covariance matrices with m-dependent random variables under the second moment condition by verifying the condition of Theorem 1.1 in Bai and Zhou (2008 Bai , Z. D. , Zhou , W. , ( 2008 ). Large sample covariance matrices without independence structure in columns . Statist. Sinica 2 : 425 – 443 . [Google Scholar]).     

New results on the convergence of random matrices

This paper extends the previous convergence results in Cerqueti and Costantini (2008) to a more general case using larger normed set of functions. In this regard, the weight-based convergence of the random matrices and their generalized eigenvalues is obtained under less restrictive requirements for the weights.     

A correlation-theoretical interpretation of schur analysis

This paper is heavily leaned on the author's recent investigations concerning SCHUR analysis of non-negative Hermitian block matrices. The parameters of the matrix balls and the triangular choice scheme which describe a non-negative Hermitian block matrix will be interpreted in the framework of correlation theory