On the distribution of the second-largest latent root for certain high dimensional Wishart matrices

摘要

The distribution of the largest latent root was found by Johnstone (2001) for Wishart distributions W_(p-1)(n, ∑_(p-1)) with large dimension p - 1, when ∑_(p-1)= I_(p-1). In this paper, we study the distribution of the second-largest latent root of the covariance matrix when ∑_p = diag(σ, 1,..., 1) with σ » 1. When N = n-1 and p are large and satisfy N/(p - 1) → γ~* ≥ 1, we shall obtain the approximate distribution of the second-largest latent root, and verify the accuracy of the approximate distribution via a simulation study.