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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Exact probabilities of correct classifications for uncorrelated repeated measurements from elliptically contoured distributions

Euclidean distance-based classification rules are derived within a certain nonclassical linear model approach and applied to elliptically contoured samples having a density generating function g. Then a geometric measure theoretical method to evaluate exact probabilities of correct classification for multivariate uncorrelated feature vectors is developed. When doing this one has to measure suitably defined sets with certain standardized measures. The geometric key point is that the intersection percentage functions of the areas under investigation coincide with those of certain parabolic cylinder type sets. The intersection percentage functions of the latter sets can be described as threefold integrals. It turns out that these intersection percentage functions yield simultaneously geometric representation formulae for the doubly noncentral g-generalized F-distributions. Hence, we get beyond new formulae for evaluating probabilities of correct classification new geometric representation formulae for the doubly noncentral g-generalized F-distributions. A numerical study concerning several aspects of evaluating both probabilities of correct classification and values of the doubly noncentral g-generalized F-distributions demonstrates the advantageous computational properties of the present new approach. This impression will be supported by comparison with the literature.It is shown that probabilities of correct classification depend on the parameters of the underlying sample distribution through a certain well-defined set of secondary parameters. If the underlying parameters are unknown, we propose to estimate probabilities of correct classification.     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.     

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.     

Some properties of invariant polynomials with matrix arguments and their applications in econometrics

Summary Further properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes thek-class estimator in econometrics.     

On the asymptotic joint distributions of certain functions of the eigenvalues of four random matrices

In this paper, the authors obtained asymptotic expressions for the joint distributions of certain functions of the eigenvalues of the Wishart matrix, correlation matrix, MANOVA matrix and canonical correlation matrix when the population roots have multiplicity.     

Estimation of the eigenvalues of noncentrality parameter in matrix variate noncentral beta distribution

We consider the problem of estimating the eigenvalues of noncentrality parameter matrix in a matrix variate noncentral beta distribution, also known as multivariate noncentral F distribution. A decision theoretic approach is taken with square error as the loss function. We propose two types of new estimators and show their superior performance theoretically as well as numerically.     

Wishartness and independence of matrix quadratic forms in a normal random matrix

Let Y be an n×p multivariate normal random matrix with general covariance Σ_Y. The general covariance Σ_Y of Y means that the collection of all np elements in Y has an arbitrary np×np covariance matrix. A set of general, succinct and verifiable necessary and sufficient conditions is established for matrix quadratic forms Y^′W_iY's with the symmetric W_i's to be an independent family of random matrices distributed as Wishart distributions. Moreover, a set of general necessary and sufficient conditions is obtained for matrix quadratic forms Y^′W_iY's to be an independent family of random matrices distributed as noncentral Wishart distributions. Some usual versions of Cochran's theorem are presented as the special cases of these results.     

On the power of Wilks' U-test for MANOVA

Let V₁,…, V_m, W₁,…, W_n be independent p × 1 random vectors having multivariate normal distributions with common nonsingular covariance matrix Σ and with EW_α = 0, α = 1,…, n. In this canonical form of the multivariate linear model, the problem is to test H: EV_αazμ_α = 0, α = 1,…, m vs K: not H. It is shown that when the rank of the noncentrality matrix (μ₁…μ_m) Σ^?1 (μ₁…μ_m) is one, the power of Wilks' U-test (the likelihood ratio test) strictly decreases with the dimension p and the hypothesis degrees of freedom m. This generalizes results known for the noncentral F-test in the univariate case.     

Some tests for the covariance matrix with fewer observations than the dimension under non-normality

This article analyzes whether some existing tests for the p×p covariance matrix Σ of the N independent identically distributed observation vectors work under non-normality. We focus on three hypotheses testing problems: (1) testing for sphericity, that is, the covariance matrix Σ is proportional to an identity matrix I_p; (2) the covariance matrix Σ is an identity matrix I_p; and (3) the covariance matrix is a diagonal matrix. It is shown that the tests proposed by Srivastava (2005) for the above three problems are robust under the non-normality assumption made in this article irrespective of whether N≤p or N≥p, but (N,p)→∞, and N/p may go to zero or infinity. Results are asymptotic and it may be noted that they may not hold for finite (N,p).     

On some distribution problems in Manova and discriminant analysis

Asymptotic expansions, valid for large error degrees of freedom, are given for the multivariate noncentral F distribution and for the distribution of latent roots in MANOVA and discriminant analysis. The asymptotic results are expressed in terms of elementary functions which are easy to compute and the results of some numerical work are included. The Bartlett test of the null hypothesis that some of the noncentrality parameters in discriminant analysis are zero is also briefly discussed.     

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.     

Lower bounds for the distributions of certain multivariate test statistics

A lower (upper) bound is given for the distribution of each d_j, j = k + 1, …, p (j = 1, …, s), the jth latent root of AB^?1, where A and B are independent noncentral and central Wishart matrices having $\text{W}{}_{\mathrm{p}}\text{(q, Σ; Ω)}$ with rank $\text{(Ω) ≤ k = p ? s}$ and W_p(n, Σ), respectively. Similar bound are also given for the distributions of noncentral means and canonical correlations. The results are applied to obtain lower bounds for the null distributions of some multivariate test statistics in Tintner's model, MANOVA and canonical analysis.