Approximation of the power functions of Roy's largest root test under general spiked alternatives

Roy's largest root is a common test statistic in multivariate analysis, statistical signal processing and related fields. According to Anderson [An Introduction to Multivariate Statistical Analysis, 3rd edn. (Wiley, New York, 2003)], it is numerically known that compared with the other three tests of linear hypotheses, Roy's largest root test has the highest power under rank-one alternatives. Therefore, it is important to study the asymptotic distribution of the largest root under rank-one alternatives to obtain an estimation of the power. To the best of our knowledge, no one had solved the problem until Johnstone and Nadler [Roy's largest root test under rank-one alternatives, Biometrika 104(1) (2017) 181-193] presented a tractable approximation of the distribution of Roy's largest root test where the alternative is of rank one and the variance of the noise tends to zero. It is natural to ask how Roy's largest root test performs under other alternatives, for example, rank-finite alternatives. Therefore, we are more interested in the power estimates of Roy's largest root test under wider alternatives, namely, whether its power is still higher than that of the other three tests of linear hypotheses. In fact, the distribution of the largest root under rank-finite alternatives can be characterized as the distribution of the largest sample eigenvalue among several spiked models. In this paper, we employ the asymptotic results of the spiked eigenvalues derived by Bai and Yao [Central limit theorems for eigenvalues in a spiked population model, Ann. Inst. Henri. Poincare Probab. Statist. 44(3) (2008) 447-474; On sample eigenvalues in a generalized spiked population model, J. Multivar. Anal. 106 (2012) 167-177] and Wang and Yao [Extreme eigenvalues of large-dimensional spiked Fisher matrices with application, Ann. Statist. 45 (2017) 415-460] to obtain the approximate power of Roy's largest root test under a high-dimensional setting; more importantly, our results are distribution-free and applicable to cases involving rank-finite alternatives.