Noncentral matrix quadratic forms of the skew elliptical variables

In this paper, the noncentral matrix quadratic forms of the skew elliptical variables are studied. A family of the matrix variate noncentral generalized Dirichlet distributions is introduced as the extension of the noncentral Wishart distributions, the Dirichlet distributions and the noncentral generalized Dirichlet distributions. Main distributional properties are investigated. These include probability density and closure property under linear transformation and marginalization, the joint distribution of the sub-matrices of the matrix quadratic forms in the skew elliptical variables and the moment generating functions and Bartlett's decomposition of the matrix quadratic forms in the skew normal variables. Two versions of the noncentral Cochran's Theorem for the matrix variate skew normal distributions are obtained, providing sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the matrix variate noncentral generalized Dirichlet distributions. Applications include the properties of the least squares estimation in multivariate linear model and the robustness property of the Wilk's likelihood ratio statistic in the family of the matrix variate skew elliptical distributions.     

Exact nonnull distribution of Wilks’ statistic: The ratio and product of independent components

The study of the noncentral matrix variate beta type distributions has been sidelined because the final expressions for the densities depend on an integral that has not been resolved in an explicit way. We derive an exact expression for the nonnull distribution of Wilks’ statistic and precise expressions for the densities of the ratio and product of two independent components of matrix variates where one matrix variate has the noncentral matrix variate beta type I distribution and the other has the matrix variate beta type I distribution. We provide the expressions for the densities of the determinant of the ratio and the product of these two components. These distributions play a fundamental role in various areas of statistics, for example in the criteria proposed by Wilks.     

Estimation of the eigenvalues of noncentrality parameter matrix in noncentral Wishart distribution

We consider the problem of estimating the eigenvalues of noncentrality parameter matrix in noncentral Wishart distribution when the scale parameter is known. A decision theoretic approach is taken with squared error as the loss function. We propose two new estimators and show their superior performance to an usual estimator theoretically and numerically.     

Nonnull distribution of likelihood ratio criterion for reality of covariance matrix

In this paper the distribution of the likelihood ratio test for testing the reality of the covariance matrix of a complex multivariate normal distribution is investigated. Some simplifications in the noncentral distribution are made and the noncentral distribution is derived for the special case where the rank of the noncentrality matrix is two. In the null case exact expressions for the distribution are given up to p = 6, and percentage points are tabulated. These percentage points were compared with percentage points derived from an asymptotic expansion of the distribution, and the accuracy of the approximation was found to be sufficient for several practical situations.     

Normal theory likelihood ratio statistic for mean and covariance structure analysis under alternative hypotheses

The normal distribution based likelihood ratio (LR) statistic is widely used in structural equation modeling. Under a sequence of local alternative hypotheses, this statistic has been shown to asymptotically follow a noncentral chi-square distribution. In practice, the population mean vector and covariance matrix as well as the model and sample size are always fixed. It is hard to justify the validity of the noncentral chi-square distribution for the resulting LR statistic even when data are normally distributed and sample size is large. By extending results in the literature, this paper develops normal distributions to describe the behavior of the LR statistic for mean and covariance structure analysis. A sequence of local alternative hypotheses is not necessary for the proposed distributions to be asymptotically valid. When the effect size is medium and above or when the model is not trivially misspecified, empirical results indicate that a refined normal distribution describes the behavior of the LR statistic better than the commonly used noncentral chi-square distribution, as measured by the Kolmogorov-Smirnov distance. Quantile-quantile plots are also provided to better understand the different distributions.     

The asymptotic covariance matrix of sample correlation coefficients under general conditions

A general matrix expression for the asymptotic covariance matrix of correlation coefficients is derived. It is applicable when the data are drawn from any distribution with finite fourth order moments. The result is specialized to the cases where the data have a distribution from the elliptical class and where the sample covariance matrix has a noncentral Wishart distribution.     

矩阵非中心Г－分布

本文定义了矩阵形式的非中心－分布。它将常见的一元Ｇａｍｍａ分布，非中心Ｘ２－分布，和多元分析中的中心则Ｗｉｓｈａｒｔ分布，非中心Ｆ－分布等纳入一个整体，进而推导了它的特征函数与特征根的分布密度函数，为一些检验奠定了基础．     

Noncentral elliptical configuration density

The noncentral configuration density, derived under an elliptical model, generalizes and corrects the Gaussian configuration and some Pearson results. Partition theory is then used to obtain explicit configuration densities associated with matrix variate symmetric Kotz type distributions (including the normal distribution), matrix variate Pearson type VII distributions (including t and Cauchy distributions), the matrix variate symmetric Bessel distribution (including the Laplace distribution) and the matrix variate symmetric Jensen-logistic distribution.     

Asymptotic distributions for the elementary symmetric functions of two matrices under the assumption of linearity

The asymptotic distributions of the elementary symmetric functions (esf's) of the characteristic roots of a noncentral multivariate beta matrix and of the generalized correlation matrix (noncentral under the assumption of linearity) are derived.     

Moments of wishart distribution

Moments of central and noncentral Wishart distributions are obtained by differentiating their characteristic functions applying matrix derivative techniques, using a special operator which takes into account the symmetry of the matrices. As a special case, higher moments of the multivariate normal distribution are obtained, arranged automatically in a square matrix form     

Equivalent conditions for noncentral generalized Laplacianness and independence of matrix quadratic forms

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y and W be a symmetric matrix. In the present article, the property that a matrix quadratic form Y^′WY is distributed as a difference of two independent (noncentral) Wishart random matrices is called the (noncentral) generalized Laplacianness (GL). Then a set of algebraic results are obtained which will give the necessary and sufficient conditions for the (noncentral) GL of a matrix quadratic form. Further, two extensions of Cochran’s theorem concerning the (noncentral) GL and independence of a family of matrix quadratic forms are developed.     

On Wishart distribution: Some extensions

This paper proposes a unified approach that enables the Wishart distribution to be studied simultaneously in the real, complex, quaternion and octonion cases under elliptical models. In particular, the matrix multivariate elliptical distribution, the noncentral generalised Wishart distribution, the joint density of the eigenvalues and the distribution of the maximum eigenvalue are obtained for real normed division algebras.     

A power function of statistical tests dependent on elementary symmetric polynomials

Statistical tests not changed under an affine change of coordinate system are considered in the multivariate analysis. In the case of a multivariate linear model and a model using the canonical correlation analysis, these tests are functions of eigenvalues of matrices following a Wishart distribution. In this paper we prove the monotonicity property of test power functions being functions of elementary symmetric polynomials of eigenvalues of a matrix following a noncentral Wishart distribution.     

A sequence of improvements over the James-Stein estimator

In this article, we consider the problem of estimating a p-variate (p ≥ 3) normal mean vector in a decision-theoretic setup. Using a simple property of the noncentral chi-square distribution, we have produced a sequence of smooth estimators dominating the James-Stein estimator and each improved estimator is better than the previous one. It is also shown by using a technique of [5]. J. Multivariate Anal.36 121–126) that our smooth estimators can be dominated by non-smooth estimators.     

Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

The noncentral gamma distribution can be viewed as a generalization of the noncentral chi-squared distribution and it can be expressed as a mixture of a Poisson density function with a incomplete gamma function. The noncentral gamma distribution is not available in free conventional statistical programs. This paper aimed to propose an algorithm for the noncentral gamma by combining the method originally proposed by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) for the noncentral distributions with the method of inversion of the distribution function with respect to the noncentrality parameter using Newton–Raphson. The algorithms are available in pseudocode and implemented as R functions. To evaluate the accuracy and speed of computation of the algorithms implemented in R, results of the distribution function, density function, quantiles and noncentrality parameter of the noncentral incomplete gamma and its particular case, the noncentral chi-squared, were obtained for the arguments settings used by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) and Chen (J Stat Comput Simul 75(10):813–829, 2005). The implemented routines performed well and, in general, were as accurate than other approximations. The R package denoted ncg is available to download on the CRAN-R package repository http://cran.r-project.org/.     

Noncentral quadratic forms of the skew elliptical variables

In this paper the quadratic forms in the skew elliptical variables are studied. A family of the noncentral generalized Dirichlet distributions is introduced and their distribution functions and probability density functions are obtained. The moment generating functions of the quadratic forms in the skew normal variables are obtained. Sufficient and necessary conditions for the quadratic forms in the skew normal variables to have the noncentral generalized Dirichlet distributions are obtained. This leads to the noncentral Cochran's Theorem for the skew normal distribution.     

The bivariate noncentral chi-square distribution - A compound distribution approach

This paper proposes the bivariate noncentral chi-square (BNC) distribution by compounding the Poisson probabilities with the bivariate central chi-square distribution. The probability density and cumulative distribution functions of the joint distribution of the two noncentral chi-square variables are derived for arbitrary values of the correlation coefficient, degrees of freedom(s), and noncentrality parameters. Computational procedures to calculate the upper tail probabilities as well as the percentile points for selected values of the parameters, for both equal and unequal degrees of freedom, are discussed. The graphical representation of the distribution for different values of the parameters are provided. Some applications of the distribution are outlined.     

Asymptotic nonnull distributions of certain test criteria for a covariance matrix

Asymptotic expansions of the distributions of two test criteria concerning a covariance matrix are derived under local alternatives in terms of noncentral χ² variates, and under the fixed alternative in terms of standard normal distribution function and its derivatives, respectively. Some numerical comparisons with the likelihood ratio criteria are made with these test criteria.     

Singular matrix variate beta distribution

In this paper, we determine the symmetrised density of doubly noncentral singular matrix variate beta type I and II distributions under different definitions. As particular cases we obtain the noncentral singular matrix variate beta type I and II distributions and the corresponding joint density of the nonnull eigenvalues. In addition, we propose an alternative approach to find the corresponding nonsymmetrised densities. From the latter, we solve the integral proposed by Constantine [Noncentral distribution problems in multivariate analysis, Ann. Math. Statist. 34 (1963) 1270-1285] and Khatri [A note on Mitra's paper “A density free approach to the matrix variate beta distribution”, Sankhyā A 32 (1970) 311-318] and reconsidered in Farrell [Multivariate Calculation: Use of the Continuous Groups, Springer Series in Statistics, Springer, New York, 1985, p. 191], see also Díaz-García and Gutiérrez-Jáimez [Noncentral matrix variate beta distribution, Comunicación Técnica, No. I-06-06 (PE/CIMAT), Guanajuato, México, 2006, 〈http://www.cimat.mx/biblioteca/RepTec/index.html?m=2〉], for the singular and nonsingular cases.     

Distribution of quadratic forms under skew normal settings

For a class of multivariate skew normal distributions, the noncentral skew chi-square distribution is studied. The necessary and sufficient conditions under which a sequence of quadratic forms is generalized noncentral skew chi-square distributed random variables are obtained. Several examples are given to illustrate the results.