Distribution approximation of covariance matrix eigenvalues

Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix

“…Through numerical experiments, we confirmed that the approximation accuracy is sufficient when the parameter ρ k is small. The distribution approximation proposed by Tsukada and Sugiyama [16] might be useful to improve the derived approximate results when ρ k → 0 is not assumed. As a part of future work, it would be desirable to examine the robustness of the chi-square approximation to normality assumption.…”

“…where k ≤ n. Under the condition of ( 9) when k = n, Takemura and Sheena [5] proved that the distribution of individual eigenvalues for a non-singular Wishart matrix is approximated by a chi-square distribution. The improvement for that approximation, that is, when the condition listed in (9) cannot be assumed, was discussed in Tsukada and Sugiyama [16]. The following lemma was provided by Nasuda et al [17] and Takemura and Sheena [5] in the non-singular case and could be easily extended to the singular case.…”