Distribution of the Largest Eigenvalue of an Elliptical Wishart Matrix and Its Simulation

Abstract

This paper provides an alternative proof of the derivation of the distribution of the largest eigenvalue of an elliptical Wishart matrix in contrast to the result of CaroLopera et al. (2016). We show the relation between multivariate and matrix-variate t distributions. From this relation, we can generate random numbers drawn from the matrix-variate t distribution. A Monte Carlo simulation is conducted to evaluate the accuracy for the truncated distribution function of the largest eigenvalue of the elliptical Wishart matrix. Exact computation of the distribution of the smallest eigenvalue is also presented.

Generalized heterogeneous hypergeometric functions and the distribution of the largest eigenvalue of an elliptical Wishart matrix

“…In the case that the elliptical Wishart matrix is nonsingular, Shinozaki et al [18] gave the distribution of the largest eigenvalue ℓ 1 in the same manner as Theorem 5 as…”

“…The empirical distribution based on 10 6 Monte Carlo simulations is represented by F sim . The generation of X ∼ T m×n (ρ, Σ) is performed according to Theorem 3 of Shinozaki et al [18]. Table 1 indicates several percentile points of correlated and uncorrelated cases.…”

“…These results of eigenvalue distributions cover the classical results under the Gaussian model as a special case. Shinozaki et al [18] provided the alternative approach for the derivation of the exact distribution of the largest eigenvalue and conducted numerical experiments under the matrix variable t model. The largest and smallest eigenvalues of a ratio of two elliptical Wishart matrices were also given by Shinozaki and Hashiguchi [19] In the case of a singular Wishart matrix, Uhlig [22] provided useful Jacobians for the transformation of singular matrices and its density.…”

Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix

“…From Lemma 2, Sugiyama (1967) derived Lemma 3 for the derivation of the exact distribution of 1 for a non-singular real Wishart matrix. Shinozaki et al (2018) gave the distribution of the largest eigenvalue under an elliptical population using Lemmas 2 and 3. In the case of a singular Wishart matrix, the exact distribution of 1 of W is given in Theorem 2.…”