Abstract: This paper aims to derive the asymptotic distributions of the spiked eigenvalues of large-dimensional spiked Fisher matrices, without imposing Gaussian assumptions or restrictive assumptions on covariance matrices. We first establish an invariance principle for the spiked eigenvalues of the Fisher matrix. That is, we show that the limiting distributions of the spiked eigenvalues are invariant over a broad range of population distributions satisfying certain conditions. Utilizing this invariance principle, we establish a central limit theorem (CLT) for the spiked eigenvalues, and further explore some interesting applications by using the CLT to derive the power functions of the Roy Maximum Root test for linear hypotheses in linear models, as well as the test in signal detection. To evaluate the effectiveness of the newly proposed test, we conduct Monte Carlo simulation studies and compare its performance with existing tests.

Key words and phrases: Random matrix theory, Roy Maximum Root test, spiked model, two-sample covariance problem.