Inference on testing the number of spikes in a high-dimensional generalized spiked Fisher matrix

The spiked Fisher matrix is a significant topic for two-sample problems in multivariate statistical inference. This paper is dedicated to testing the number of spikes in a high-dimensional generalized spiked Fisher matrix that relaxes the Gaussian population assumption and the diagonal constraints on the population covariance matrices. First, we propose a general test statistic predicated on partial linear spectral statistics to test the number of spikes, then establish the central limit theorem (CLT) for this statistic under the null hypothesis. Second, we apply the CLT to address two statistical problems: variable selection in high-dimensional linear regression and change point detection. For each test problem, we construct new statistics and derive their asymptotic distributions under the null hypothesis. Finally, simulations and empirical analysis are conducted to demonstrate the remarkable effectiveness and generality of our proposed methods across various scenarios.