Abstract

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.

摘要

难以有效地计算 MANOVA 矩阵的最大特征值的零分布阻碍了罗伊最大根检验 (RLRT) 的更广泛适用性，尽管它是在六年前提出的。Johnstone、Butler 以及 Paige 和 Chiani 最近取得的进展大大简化了 RLRT 临界值的近似和精确计算。当数据集是高维 (HD) 时，由于截断误差，Chiani 的精确计算数值算法可能无法给出可靠的结果，而 Johnstone 通过 Tracy-Widom 分布的近似方法可能会给出一个很好的近似值。在本文中，我们进行了比较研究，以研究精确方法在哪个区域给出了可靠的数值，以及 Johnstone 方法在哪个区域给出了高质量的近似值。我们制定建议以告知 RLRT 从业者。我们还在高维环境中进行模拟研究，以检查 RLRT 对人口正态性假设的稳健性。我们的研究支持 RLRT 对 HD 中的非正态性的鲁棒性。