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.     

Asymptotic distribution of Wishart matrix for block-wise dispersion of population eigenvalues

This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically generate two Wishart matrices and one normally distributed random matrix, which are mutually independent. For a family of orthogonally equivariant estimators, we calculate the asymptotic risks with respect to the entropy or the quadratic loss function and derive the asymptotically best estimator among the family. We numerically show (1) the convergence in both the distributions and the risks are quick enough for a practical use, (2) the asymptotically best estimator is robust against the deviation of the population eigenvalues from the block-wise infinite dispersion.     

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.     

On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008.     

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.     

Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data

In this article, the Stein-Haff identity is established for a singular Wishart distribution with a positive definite mean matrix but with the dimension larger than the degrees of freedom. This identity is then used to obtain estimators of the precision matrix improving on the estimator based on the Moore-Penrose inverse of the Wishart matrix under the Efron-Morris loss function and its variants. Ridge-type empirical Bayes estimators of the precision matrix are also given and their dominance properties over the usual one are shown using this identity. Finally, these precision estimators are used in a quadratic discriminant rule, and it is shown through simulation that discriminant methods based on the ridge-type empirical Bayes estimators provide higher correct classification rates.     

Estimating the covariance matrix: a new approach

In this paper, we consider the problem of estimating the covariance matrix and the generalized variance when the observations follow a nonsingular multivariate normal distribution with unknown mean. A new method is presented to obtain a truncated estimator that utilizes the information available in the sample mean matrix and dominates the James-Stein minimax estimator. Several scale equivariant minimax estimators are also given. This method is then applied to obtain new truncated and improved estimators of the generalized variance; it also provides a new proof to the results of Shorrock and Zidek (Ann. Statist. 4 (1976) 629) and Sinha (J. Multivariate Anal. 6 (1976) 617).     

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n→∞ with p/n→c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.     

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.     

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.     

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.     

关于对称矩阵随机变换方差的证明

应用正交变换将对称矩阵对角化,基于随机向量正交变换后独立性的不变性及矩阵迹相关性质,给出一个关于对称矩阵经随机变换后方差的证明,并将该结论推广到更一般情形。     

The characterizations of a semicircle law by the certain freeness in a C*-probability space

In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper, we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables, which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system. Our proof is more direct and straightforward one. Received: 12 February 1997 / Revised version: 16 June 1998     

有限域上随机变量联合分布及二阶矩的分解与应用

本文给出了有限域上随机变量联合概率和二阶矩的分解公式，给出了有限域上随机变量相互独立的谱刻划，应用上述结果，建立了在进行频次分析时，对有限域上随机向量构造的Χ平方统计量与该随机向量坐标函数的非零线性组合的Χ平方统计量之间的内在联系，给出了有限域上相关免疫函数谱特征的新证明，建立了有限域上多输出函数的差分分布与其广义Chrestenson循环谱之间的内在联系，建立了多输出函数的平衡性其差分分布之间的内在联系。     

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.     

A note about measures and Jacobians of singular random matrices

This paper explains the differences between the densities and the Jacobians of the transforms of the same singular random matrices treated by several authors. Some comments on the results proposed by Srivastava [Singular Wishart and multivariate beta distributions, Ann. Statist. 31 (2003) 1537-1560] are presented. Definitions about a measure with respect to which a singular random matrix possesses a density are proposed. Finally two Jacobians of certain transforms under any of those measures are found.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

A trivariate chi-squared distribution derived from the complex Wishart distribution

The joint density for a particular trivariate chi-squared distribution given by the diagonal elements of a complex Wishart matrix is derived. This distribution has applications in the processing of multilook synthetic aperture radar data. The expression for the density is in the form of an infinite series that converges rapidly and is simple and fast to compute. The expression is shown to reduce to known forms for a number of special cases and is validated by simulation. The characteristic function is also derived and used to relate joint moments of the trivariate distribution to the parameters of the density function.     

Multivariate generalized S-estimators

In this paper we introduce generalized S-estimators for the multivariate regression model. This class of estimators combines high robustness and high efficiency. They are defined by minimizing the determinant of a robust estimator of the scatter matrix of differences of residuals. In the special case of a multivariate location model, the generalized S-estimator has the important independence property, and can be used for high breakdown estimation in independent component analysis. Robustness properties of the estimators are investigated by deriving their breakdown point and the influence function. We also study the efficiency of the estimators, both asymptotically and at finite samples. To obtain inference for the regression parameters, we discuss the fast and robust bootstrap for multivariate generalized S-estimators. The method is illustrated on a real data example.     

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.