Roy's Largest Root Test Under Rank-One Alternatives

Roy's largest root is a common test statistic in multivariate analysis, statistical signal processing and allied fields. Despite its ubiquity, provision of accurate and tractable approximations to its distribution under the alternative has been a longstanding open problem. Assuming Gaussian observations and a rank one alternative, or concentrated non-centrality, we derive simple yet accurate approximations for the most common low-dimensional settings. These include signal detection in noise, multiple response regression, multivariate analysis of variance and canonical correlation analysis. A small noise perturbation approach, perhaps underused in statistics, leads to simple combinations of standard univariate distributions, such as central and non-central $\chi^2$ and $F$. Our results allow approximate power and sample size calculations for Roy's test for rank one effects, which is precisely where it is most powerful.