On the evaluation of some distributions that arise in simultaneous tests for the equality of the latent roots of the covariance matrix

In this paper, the authors consider the evaluation of the distribution functions of the ratios of the intermediate roots to the trace of the real Wishart matrix as well as the ratios of the individual roots to the trace of the complex Wishart matrix. In addition, the authors consider the evaluation of the distribution functions of the ratios of the extreme roots of the Wishart matrix in the real and complex cases. Some applications and tables of the above distributions are also given.     

A characterization of Wishart processes and Wishart distributions

A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices.     

Craig-Sakamoto's theorem for the Wishart distributions on symmetric cones

A version of Craig-Sakamoto's theorem says essentially that ifX is aN(O,I _n ) Gaussian random variable in ⁿ, and ifA andB are (n, n) symmetric matrices, thenXAX andXBX (or traces ofAXX andBXX) are independent random variables if and only ifAB=0. As observed in 1951, by Ogasawara and Takahashi, this result can be extended to the case whereXX is replaced by a Wishart random variable. Many properties of the ordinary Wishart distributions have recently been extended to the Wishart distributions on the symmetric cone generated by a Euclidean Jordan algebraE. Similarly, we generalize there the version of Craig's theorem given by Ogasawara and Takahashi. We prove that ifa andb are inE and ifW is Wishart distributed, then Tracea.W and Traceb.W are independent if and only ifa.b=0 anda.(b.x)=b.(a.x) for allx inE, where the. indicates Jordan product.Partially supported by NATO grant 92.13.47.     

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.     

On Transformations and Determinants of Wishart Variables on Symmetric Cones

Let x and y be independent Wishart random variables on a simple Jordan algebra V. If c is a given idempotent of V, write for the decomposition of x in where V(c,) equals the set of v such that cv=v. In this paper we compute E(det(ax+by)) and some generalizations of it (Theorems 5 and 6). We give the joint distribution of (x ₁, x ₁₂, y ₀) where and P is the quadratic representation in V. In statistics, if x is a real positive definite matrix divided into the blocks x ₁₁, x ₁₂, x ₂₁, x ₂₂, then y ₀ is equal to . We also compute the joint distribution of the eigenvalues of x (Theorem 9). These results have been known only when V is the algebra of Hermitian matrices with entries in the real or the complex field. To obtain our results, we need to prove several new results on determinants in Jordan algebras. They include in particular extensions of some classical parts of linear algebra like Leibnitz's determinant formula (Proposition 2) or Schur's complement (Eqs. (3.3) and (3.6)).     

Asymptotic expansions of the distributions of the latent roots and the latent vector of the Wishart and multivariate F matrices

Asymptotic expansions of the joint distributions of the latent roots of the Wishart matrix and multivariate F matrix are obtained for large degrees of freedom when the population latent roots have arbitrary multiplicity. Asymptotic expansions of the distributions of the latent vectors of the above matrices are also derived when the corresponding population root is simple. The effect of normalizations of the vector is examined.     

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n^?2, for the ₀F₀ function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.     

Improved approximations to distributions of the largest and the smallest latent roots of a wishart matrix

Summary Normalizing transformations of the largest and the smallest latent roots of a sample covariance matrix in a normal sample are obtained, when the corresponding population roots are simple. Using our results, confidence intervals for population roots may easily be constructed. Some numerical comparisons of the resulting approximations are made in a bivariate case, based on exact values of the probability integral of latent roots.     

Bivariate gamma distributions, sums and ratios

Exact distributions of R = X +Y and W = X/(X +Y ) and the corresponding moment properties are derived when X and Y follow five flexible bivariate gamma distributions. The expressions turn out to involve several special functions.     

On the non-commutative fractional Wishart process

We investigate the process of eigenvalues of a fractional Wishart process defined by $N=B^{?}B$, where B is the matrix fractional Brownian motion recently studied in [18]. Using stochastic calculus with respect to the Young integral we show that, with probability one, the eigenvalues do not collide at any time. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in (1/2,1)$, we derive a stochastic differential equation in the Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally, a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the non-commutative fractional Wishart process, which constitutes the family of fractional dilations of the free Poisson distribution.     

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.     

关于矩阵迹的一些不等式

对由Bellman不等式推导的两个关于实正定对称矩阵迹的不等式进行了进一步的推广,并得到了一系列的关于矩阵迹的不等式。     

An identity for the Wishart distribution with applications

Let S_p×p have a Wishart distribution with unknown matrix Σ and k degrees of freedom. For a matrix T(S) and a scalar h(S), an identity is obtained for E_∑tr[h(S)T∑⁻¹]. Two applications are given. The first provides product moments and related formulae for the Wishart distribution. Higher moments involving S can be generated recursively. The second application concerns good estimators of ∑ and ∑⁻¹. In particular, identities for several risk functions are obtained, and estimators of ∑ (∑⁻¹) are described which dominate aS(bS⁻¹), a ≤ 1/k (b ≤ k − p − 1). [3] Ann. Statist. 7 No. 5; (1980) Ann. Statist. 8 used special cases of the identity to find unbiased risk estimators. These are unobtainable in closed form for certain natural loss functions. In this paper, we treat these case as well. The dominance results provide a unified theory for the estimation of ∑ and ∑⁻¹.     

Asymptotic expansions for the distributions of latent roots ofS h S e −1 and of certain test statistics in MANOVA

S _e andS _n are independent central and noncentral Wishart matrices having Wishart distributionsW _p (n _e , Σ) andW _p (n _h , Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS _h S _e ⁻¹ and of certain test statistics in MANOVA under the assumption thatn=n _e +n _h becomes large with a fixed ration _e ∶n _h =e∶h(e+h=1,e>0,h>0) andΩ=O(n).     

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.     

Asymptotic expansions for the distributions of functions of a correlation matrix

This paper deals with asymptotic expansions for the non-null distributions of certain test statistics concerning a correlation matrix in a multivariate normal distribution. For this purpose an asymptotic expansion is given for the distribution of a function of the sample correlation matrix. As special cases of the resulting expansion, asymptotic expansions for the distributions of the sample correlation coefficient, Fisher's z-transformation and arcsine transformation are also given.     

On roots of random polynomials

We study the distribution of the complex roots of random polynomials of degree with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.

     

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.     

The expected values of invariant polynomials with matrix argument of elliptical distributions

ThisresearchissupportedbytheChineseAcademyofSciences.1.IntroductionTherearemailypublicatiollsonmatrixvariatedistributions(ormatrixdistributionforsimplicity),inparticular,abollttheirexpectedvaluesofzonalpolynomialsoftheirquadraticforms(of.[12],[151,[171and…