Exact and approximate computation of critical values of the largest root test in high dimension

The difficulty to efficiently compute the null distribution of the largest eigenvalue of a MANOVA matrix has hindered the wider applicability of Roy's Largest Root Test (RLRT) though it was proposed over six decades ago. Recent progress made by Johnstone, Butler and Paige and Chiani has greatly simplified the approximate and exact computation of the critical values of RLRT. When datasets are high dimensional (HD), Chiani's numerical algorithm of exact computation may not give reliable results due to truncation error, and Johnstone's approximation method via Tracy-Widom distribution is likely to give a good approximation. In this paper, we conduct comparative studies to study in which region the exact method gives reliable numerical values, and in which region Johnstone's method gives a good quality approximation. We formulate recommendations to inform practitioners of RLRT. We also conduct simulation studies in the high dimensional setting to examine the robustness of RLRT against normality assumption in populations. Our study provides support of RLRT robustness against non-normality in HD.