Asymptotic Distribution of Wishart Matrix for Block-wise Dispersion of Population Eigenvalues

In this paper we derive some new and practical results on testing and interval estimation problems for the population eigenvalues of a Wishart matrix based on the asymptotic theory for block-wise infinite dispersion of the population eigenvalues. This new type of asymptotic theory has been developed by the present authors in Takemura and Sheena (2005) and Sheena and Takemura (2007a,b) and in these papers it was applied to point estimation problem of population covariance matr...

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., "eigen-inference"). Results found in the literature establish the asymptotic normality of the fluctuatio...

The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distribution with respect to its central counterpart. However, no source has yet developed explicit local approximations for the (central) Wi...

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom ...

In the present work, eigenvalue distributions defined by a random rectangular matrix whose components are neither independently nor identically distributed are analyzed using replica analysis and belief propagation. In particular, we consider the case in which the components are independently but not identically distributed; for example, only the components in each row or in each column may be {identically distributed}. We also consider the more general case in which the comp...

This paper deals with matrix-variate distributions, from Wishart to Inverse Elliptical Wishart distributions over the set of symmetric definite positive matrices. Similar to the multivariate scenario, (Inverse) Elliptical Wishart distributions form a vast and general family of distributions, encompassing, for instance, Wishart or $t$-Wishart ones. The first objective of this study is to present a unified overview of Wishart, Inverse Wishart, Elliptical Wishart, and Inverse El...

In this paper, we derive the explicit series expansion of the eigenvalue distribution of various models, namely the case of non-central Wishart distributions, as well as correlated zero mean Wishart distributions. The tools used extend those of the free probability framework, which have been quite successful for high dimensional statistical inference (when the size of the matrices tends to infinity), also known as free deconvolution. This contribution focuses on the finite Ga...

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hyper-geometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equ...

Let X be a n*p matrix and l_1 the largest eigenvalue of the covariance matrix X^{*}*X. The "null case" where X_{i,j} are independent Normal(0,1) is of particular interest for principal component analysis. For this model, when n, p tend to infinity and n/p tends to gamma in (0,\infty), it was shown in Johnstone (2001) that l_1, properly centered and scaled, converges to the Tracy-Widom law. We show that with the same centering and scaling, the result is true even when p/n or n...

We consider Bayesian inference on the spiked eigenstructures of high-dimensional covariance matrices; specifically, we focus on estimating the eigenvalues and corresponding eigenvectors of high-dimensional covariance matrices in which a few eigenvalues are significantly larger than the rest. We impose an inverse-Wishart prior distribution on the unknown covariance matrix and derive the posterior distributions of the eigenvalues and eigenvectors by transforming the posterior d...