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.     

Characterizations of the beta distribution on symmetric matrices

This paper extends to the beta-Wishart distribution on symmetric matrices, two characterizations of the beta distributions on , due to Seshadri and Wesolowski and based on some properties of constancy regression.     

Quadratic forms of a matric-t variate

Summary Crowther [2] studied the distribution of a quadratic form in a matrix normal variate. This, in some sense, is extended by De Waal [4]. They represented the density function of this quadratic form in terms of generalized Hayakawa polynomials. Application of some specific results of these authors facilitates the derivation of distributions of quadratic forms of the matric-t variate. Attention is also given to the distributions of the characteristic roots and the trace of this quadratic matrix. Special cases are considered and some useful integrals are formulated. Financially supported by the CSIR and the University of the Orange Free State     

The two-dimensional beta binomial distribution

We introduce the two-dimensional beta binomial distribution to describe correlated counts data. Properties of the distribution are explored and in an example the distribution is used to describe a set of orthodontic data.     

Approximating by the Wishart distribution

Approximations of density functions are considered in the multivariate case. The results are presented with the help of matrix derivatives, powers of Kronecker products and Taylor expansions of functions with matrix argument. In particular, an approximation by the Wishart distribution is discussed. It is shown that in many situations the distributions should be centred. The results are applied to the approximation of the distribution of the sample covariance matrix and to the distribution of the non-central Wishart distribution.     

On the distribution singular values of Toeplitz matrices

We prove a second order formula concerning distribution of singular values of Toeplitz matrices in some cases when conditions of the H. Widom Theorem are not satisfied.

     

An alternative method for computing mean and covariance matrix of some multivariate distributions

Computing the mean and covariance matrix of some multivariate distributions, in particular, multivariate normal distribution and Wishart distribution are considered in this article. It involves a matrix transformation of the normal random vector into a random vector whose components are independent normal random variables, and then integrating univariate integrals for computing the mean and covariance matrix of a multivariate normal distribution. Moment generating function technique is used for computing the mean and covariances between the elements of a Wishart matrix. In this article, an alternative method that uses matrix differentiation and differentiation of the determinant of a matrix is presented. This method does not involve any integration.     

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.     

Properties of the complex bimatrix variate beta distribution

In this article, we derive several properties such as marginal distribution, moments involving zonal polynomials, and asymptotic expansion of the complex bimatrix variate beta type 1 distribution introduced by D?´az-Garc?´a and Gutiérrez Jáimez [José A. D?´az-Garc?´a, Ramón Gutiérrez Jáimez, Complex bimatrix variate generalised beta distributions, Linear Algebra Appl. 432 (2010) 571-582]. We also derive distributions of several matrix valued functions of random matrices jointly distributed as complex bimatrix variate beta type 1.     

On the structure of the Wishart distribution

In this paper it is shown that every nonnegative definite symmetric random matrix with independent diagonal elements and at least one nondegenerate nondiagonal element has a noninfinitely divisible distribution. Using this result it is established that every Wishart distribution W_p(k, Σ, M) with both p and rank (Σ) ≥ 2 is noninfinitely divisible. The paper also establishes that any Wishart matrix having distribution W_p(k, Σ, 0) has the joint distribution of its elements in the rth row and rth column to be infinitely divisible for every r = 1,2,…,p.     

Maximum principle for optimal control of distributed-parameter systems with singular mass matrix

A maximum principle for the open-loop optimal control of a vibrating system relative to a given convex index of performance is investigated. Though maximum principles have been studied by many people (see, e.g., Refs. 1–5), the principle derived in this paper is of particular use for control problems involving mechanical structures. The state variable satisfies general initial conditions as well as a self-adjoint system of partial differential equations together with a homogeneous system of boundary conditions. The mass matrix is diagonal, constant, and singular, and the viscous damping matrix is diagonal. The maximum principle relates the optimal control with the solution of the homogeneous adjoint equation in which terminal conditions are prescribed in terms of the terminal values of the optimal state variable. An application of this theory to a structural vibrating system is given in a companion paper (Ref. 6).     

Some new properties of the Riesz probability distribution

The Riesz probability distribution on any symmetric cone and, in particular, on the cone of positive definite symmetric matrices represents an important generalization of the Wishart and of the matrix gamma distributions containing them as particular examples. The present paper is a continuation of the investigation of the properties of this probability distribution. We first establish a property of invariance of this probability distributions by a subgroup of the orthogonal group. We then show that the Pierce components of a Riesz random variable are independent, and we determine their probability distributions. Some moments and some useful expectations related to the Riesz probability distribution are also calculated. Copyright © 2017 John Wiley & Sons, Ltd.     

On the projections of the Dirichlet probability distribution on symmetric cones

In this paper, we introduce the Riesz-Dirichlet distribution on a symmetric cone as an extension of the Dirichlet distribution defined by the Wishart distribution. We also show that some projections of these distributions related to the Pierce decomposition are also Dirichlet.     

Minimax and admissible minimax estimators of the mean of a multivariate normal distribution for unknown covariance matrix

Let X be a p-variate (p ≥ 3) vector normally distributed with mean μ and covariance Σ, and let A be a p × p random matrix distributed independent of X, according to the Wishart distribution W(n, Σ). For estimating μ, we consider estimators of the form δ = δ(X, A). We obtain families of Bayes, minimax and admissible minimax estimators with respect to the quadratic loss function (δ ? μ)′ Σ^?1(δ ? μ) where Σ is unknown. This paper extends previous results of the author [1], given for the case in which the covariance matrix of the distribution is of the form σ²I, where σ is known.     

Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules

We have studied the asymptotics of two special two-matrix hypergeometric functions. The validity of the asymptotic expressions for these functions is seen in several selected numerical comparisons between the exact and asymptotic results. These hypergeometric functions find applications in configuration statistics of macromolecules as well as multivariate statistics.This work was supported by grant DE-FG06-84ER45123 from the Department of Energy, U.S.A.     

On the Power Function of the Likelihood Ratio Test for MANOVA

We prove that the power function of the likelihood ratio test for MANOVA attains its minimum when the rank of the location parameter matrix Θ decreases from s to 1. This provides a theoretical justification of a result that is known in the literature based only on numerical studies.     

Optimality of singular vector perturbation under maximum norm

Fan, Wang, and Zhong estimate the difference between the singular vectors of a matrix and those of a perturbed matrix in terms of the maximum norm. Their estimations are used effectively to establish the asymptotic properties of robust covariance estimators (see Journal of Machine Learning Research, 2018;18:1-42). In this paper, we give the corresponding lower bound estimates, which show Fan-Wang-Zhong's estimations optimal.     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.     

离散广义系统的区间矩阵平稳振荡

利用范数理论和代数方法,研究了离散广义系统的区间平稳振荡问题．给出了两种区间矩阵平稳振荡存在的充分条件．提供了可行性的算例．