Singular random matrix decompositions: distributions

Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and pseudo-Wishart generalized singular and non-singular distributions. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.     

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.     

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.     

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.     

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.     

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.     

Wishart ratios with dependent structure: New members of the bimatrix beta type IV

In multivariate statistics under normality, the problems of interest are random covariance matrices (known as Wishart matrices) and “ratios” of Wishart matrices that arise in multivariate analysis of variance (MANOVA) (see 24). The bimatrix variate beta type IV distribution (also known in the literature as bimatrix variate generalised beta; matrix variate generalization of a bivariate beta type I) arises from “ratios” of Wishart matrices. In this paper, we add a further independent Wishart random variate to the “denominator” of one of the ratios; this results in deriving the exact expression for the density function of the bimatrix variate extended beta type IV distribution. The latter leads to the proposal of the bimatrix variate extended F distribution. Some interesting characteristics of these newly introduced bimatrix distributions are explored. Lastly, we focus on the bivariate extended beta type IV distribution (that is an extension of bivariate Jones’ beta) with emphasis on P(X₁<X₂) where X₁ is the random stress variate and X₂ is the random strength variate.     

Complex bimatrix variate generalised beta distributions

In this paper, the study of bivariate generalised beta types I and II distributions is extended to the complex matrix variate case, for which the corresponding density functions are found. In addition, for complex bimatrix variate beta type I distributions, several basic properties, including the joint eigenvalue density and the maximum eigenvalue distribution, are studied.     

Statistical theory of shape under elliptical models and singular value decompositions

The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified BIC^∗ criterion.     

ι_1-模对称矩阵变量分布的边缘相容性(英文)

本文研究了ι1－模对称矩阵变量分布及其任意行组成的子矩阵的分布属于同一指定分布族的条件；并给出了无穷维ι1－模对称矩阵变量分布的定义．     

On Relationships Between Classical Pearson Distributions and Gauss Hypergeometric Function

In this paper, it is shown that the classical Pearson distributions and Gauss hypergeometric function satisfy a unique differential equation of hypergeometric type. Hence, they are directly related to each other. This connection leads to some new integral relations between them. For instance, two special cases of Pearson distributions, namely the generalized T distribution and Beta distribution, are considered and their direct relationships with Gauss hypergeometric function are obtained.     

Sequences of elliptical distributions and mixtures of normal distributions

Two conditions are shown under which elliptical distributions are scale mixtures of normal distributions with respect to probability distributions. The issue of finding the mixing distribution function is also considered. As a unified theoretical framework, it is also shown that any scale mixture of normal distributions is always a term of a sequence of elliptical distributions, increasing in dimension, and that all the terms of this sequence are also scale mixtures of normal distributions sharing the same mixing distribution function. Some examples are shown as applications of these concepts, showing the way of finding the mixing distribution function.     

Estimation of a covariance matrix with location: Asymptotic formulas and optimal B-robust estimators

Applying the non-singular affine transformations AZ + μ to a spherically symmetrically distributed variate Z generates the covariance-location model, indexed by the parameters AA^T and μ, consisting of so-called elliptical distributions. We develop an algebraic machinery that simplifies the derivation of influence functions and asymptotic variance-covariance matrices for equivariant estimators of Σ and μ and reveals a natural structure of Σ. In addition, optimal B-robust estimators are derived.     

Portfolio separation properties of the skew-elliptical distributions, with generalizations

The two-fund separation property of the elliptical distributions is extended to the skew-elliptical case by adding a number of funds equaling the rank of the skewness matrix. The singular extended skew-elliptical distributions are covered, as is a further generalization to the case where the set conditioned upon is not an orthant.     

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     

常利息力影响下跳扩散模型的分红问题

本文中用常值利率驱动下的经典跳扩散模型模拟保险公司的盈余过程,研究了该模型在带壁分红策略下的若干问题.首先得出破产前分红折现的高阶矩所满足的积分微分方程,并在指数分布的情况下借助合流超几何函数给出了方程的显式解.其次关于破产前聚合分红得到了一些令人满意的结果,这些结果甚至对一般的分布都成立,另外讨论了分红流的次数和额度.最后研究了指数分布时破产赤字折现期望问题.本文的部分结论深化了精算学中一些已有研究成果.     

从SPVⅡ分布生成的新的多元偏态t分布的有关性质

随机向量的t分布属于椭球等高分布族,然而,它是对称分布.在许多诸如经济学、生理学、社会学等领域中,有时回归模型中的随机误差不再满足对称性,通常表现出高度的偏态性(skewness).于是就有了偏态椭球等高分布族.本文在已有的多元偏态t分布的基础上,着重研究它的分布性质,包括线性组合分布、边缘分布、条件分布及各阶矩.     

Expressions for some hypergeometric functions of matrix argument with applications

Reasonably simple expressions are given for some hypergeometric functions when the size of the argument matrix or matrices is two. Applications of these expressions in connection with the distributions of the latent roots of a 2 × 2 Wishart matrix are also given.     

The matrix-t distribution and its applications in predictive inference

The predictive distributions of the future responses and regression matrix under the multivariate elliptically contoured distributions are derived using structural approach. The predictive distributions are obtained as matrix-t which are identical to those obtained under matrix normal and matrix-t distributions. This gives inference robustness with respect to departures from the reference case of independent sampling from the matrix normal or dependent but uncorrelated sampling from matrix-t distributions. Some successful applications of matrix-t distribution in the field of spatial prediction have been addressed.