The inverse problem of multivariate and matrix-variate skew normal distributions

In this paper, we prove that the joint distribution of random vectors Z ₁ and Z ₂ and the distribution of Z ₂ are skew normal provided that Z ₁ is skew normally distributed and Z ₂ conditioning on Z ₁ is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.     

On matrix variate skew-normal distributions

Two new families of matrix variate distributions are introduced. They are based on matrix normal distribution and yet can be used to model data involving skewness. The properties of the two families are investigated. Among others, the marginals, conditionals, stochastic representation, linear and quadratic forms are studied.     

A multivariate two-factor skew model

It is well known that many data, such as the financial or demographic data, exhibit asymmetric distributions. In recent years, researchers have concentrated their efforts to model this asymmetry. Skew normal model is one of such models that are skew and yet possess many properties of the normal model. In this paper, a new multivariate skew model is proposed, along with its statistical properties. It includes the multivariate normal distribution and multivariate skew normal distribution as special cases. The quadratic form of this random vector follows a χ² distribution. The roles of the parameters in the model are investigated using contour plots of bivariate densities.     

The risk of pretest and shrinkage estimators

In this paper, we establish the risk function of a class of estimator for the mean parameter matrix of a matrix variate normal distribution. In particular, the established result is useful in evaluating the performance of a class of shrinkage-pretest-type estimators.     

A new family of multivariate skew slash distribution

In this article, the new family of multivariate skew slash distribution is defined. According to the definition, a stochastic representation of the multivariate skew slash distribution is derived. The first four moments and measures of skewness and kurtosis of a random vector with the multivariate skew slash distribution are obtained. The distribution of quadratic forms for the multivariate skew slash distribution and the non central skew slash χ² distribution are studied. Maximum likelihood inference and real data illustration are discussed. In the end, the potential extension of multivariate skew slash distribution is discussed.     

Bivariate,multivariate, and matrix variate normal characterizations: A brief survey II

The paper entitled “Bivariate and Multivariate Normal Characterizations: A Brief Survey,” by Hamedani, which was published in 1992, covered the published characterizations of bivariate and multivariate normal (MVN) distributions from 1941 to 1991. The present work is a follow-up to the 1991/1992 survey which includes not only characterizations of the bivariate and MVN distributions, but also characterizations of the matrix variate normal distribution, which have appeared from 1991/1992 to the present.     

On the sample characterization criterion for normal distributions

Conventionally, it was shown that the underlying distribution is normal if and only if the sample mean and sample variance from a random sample are independent. This paper focusses on the normal population characterization theorem by showing that, if the joint distribution of a skew normal sample follows certain multivariate skew normal distribution, the sample mean and sample variance are still independent.     

Linear generalizations of various matric-t distributions

The concept of a matric-t variate is extended to cases where the positive (definite) part of the variate, which is usually Wishart distributed independently of the normal part, is a linear sum of positive (definite) variates with positive coefficients. These distributions and their quadratic forms are of importance i.a, for the exact solution to the multi¬variate Behrens-Fisher problem. A few useful identities con¬cerning the invariant polynomials with matrix arguments are derived     

Matrix variate Macdonald distribution

Abstract

In this article, we generalize the univariate Macdonald distribution to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of several functions of the Macdonald matrix. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials.     

Matrix variate Cauchy distribution

In this article we introduce the matrix variate Cauchy distribution. Its density function has been derived using independent random matrices having dependent normal entries. Some properties of this distribution are also studied.     

Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

Statistical estimation of nonlinear regression functions

In order to obtain the first and second moments of a matrix quadratic form under normality assumptions its moment generating function will be derived and then differentiated.

Use is being made of matrix differential calculus as developed by the author     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix‐variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large‐dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise.     

The n.s. conditions for the ration of two quadratic forms to have an f-distribution and its applications

Suppose that ξ and η be two random vectors and that (ξ^τ, η^τ have an elliptically contoured distribution or a multivariate normal distribution. In this article, we obtain some necessary and sufficient (N.S.) conditions such that the ratio of two quadratic forms, say ξ^τ Aξ and η^τ Bη(for some symmetric nonnegative matrices A and B), has an F-distribution. As applications, we extend the classical F-test to some dependent two group samples. Two cases are considered: elliptically contoured and normal distributions.     

Mean-shift outliers model in skew scale-mixtures of normal distributions

ABSTRACT

Asymmetric models have been discussed quite extensively in recent years, in situations where the normality assumption is suspected due to lack of symmetry in the data. Techniques for assessing the quality of fit and diagnostic analysis are important for model validation. This paper presents a study of the mean-shift method for the detection of outliers in regression models under skew scale-mixtures of normal distributions. Analytical solutions for the estimators of the parameters are obtained through the use of Expectation–Maximization algorithm. The observed information matrix for the calculation of standard errors is obtained for each distribution. Simulation studies and an application to the analysis of a data have been carried out, showing the efficiency of the proposed method in detecting outliers.     

Bimatrix Variate Beta Type IV Distribution: Relation to Wilks's Statistic and Bimatrix Variate Kummer-Beta Type IV Distribution

In this article, the bimatrix variate beta Type IV distribution is derived from independent Wishart distributed matrix variables. We explore specific properties of this distribution which is then used to derive the exact expressions of the densities of the product and ratio of two dependent Wilks's statistics and to define the bimatrix Kummer-beta Type IV distribution.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

A general reduction method for n–variate normal orthant probability

Let X₁,X₂,…,X_n be n normal variates with zero means, unit variances and correlation matrix {p_ij). The orthant probability is the probability that all of the X₁'s are simultaneously positive. This paper presents a general reduction method by extending the method of Childs (1967), and shows that the probability can be represented by a linear combination of some multivariate integrals of order([n/2]?1). As illustrations, we apply the proposed method to the quadrivariate and six–variate cases. Some numerical results are also given.