Eigenvalue distributions of beta-Wishart matrices

Abstract

Abstract. We derive explicit expressions for the distributions of the extreme eigenvalues of the Beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0. Our new results are particularly convenient for practical evaluation using our algorithms for the hypergeometric function of matrix argument [13], see section 5.

High dimensional statistical inference and random matrices

“…This reflects the constraining effect of eigenvalue repulsion due to the Vandermonde term in (4). The fact that the same limit arises in the single and double Wishart settings (Laguerre, Jacobi ensembles) is an instance of the universality discussed in P. Deift's paper [29] Both are based on ideas of [76] [see also [35]] assumption that the i.i.d. entries in the p × n data matrix X are Gaussian.…”

The densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix

“…We present new expressions for the density and distribution of the trace of a Wishart matrix that can be computed in time that is linear in the size of the matrix and the degree of the truncation of its series expansion. This complexity is optimal and is an exponential improvement over the previous result for the trace from [6].…”

“…Computationally, the significance of the above result is that the parameter m−1 2 β implies that the 2 F 2 hypergeometric function need only be summed over partitions of m − 1 parts instead of m. Since the number of partitions of M in not more than k parts for M k grows as O(M k ), we can compute the density of the largest eigenvalue of a Wishart matrix about O(M ) times faster than a direct differentiation of the expression for the distribution from [6]: Another advantage of the density expression of Theorem 3.1 is that it allows us to write the density of λ max as an infinite mixture of chi-squared densities. To see that, let Σ = Σ −1 /tr(Σ −1 ) and the density of λ max becomes…”

“…The eigenvalues of a Wishart matrix and its trace have long been used in multivariate statistical analysis for a variety of analyses and applications [11]. The only known expressions from [6] ((3.1) and (4.3) below), however, are in terms of infinite series of Jack functions, and in particular, the hypergeometric function of a matrix argument. These series are notoriously slow to converge and have been a computational challenge for decades despite recent progress [10].…”

A study on the empirical distribution of the scaled Hankel matrix eigenvalues

“…For recent work on the generalized eigenvalues of Hankel random matrices see Naronic article [9] . For the eigenvalue distributions of beta-Wishart matrices which is a special case of random matrix see Edelman and Plamen study [10] .…”