  Two-sample spiked model is an important issue in multivariate statistical
inference. This paper focuses on testing the number of spikes in a
high-dimensional generalized two-sample spiked model, which is free of Gaussian
population assumption and the diagonal or block-wise diagonal restriction of
population covariance matrix, and the spiked eigenvalues are not necessary
required to be bounded. In order to determine the number of spikes, we first
propose a general test, which relies on the partial linear spectral statistics.
We establish its asymptotic normality under the null hypothesis. Then we apply
the conclusion to two statistical problem, variable selection in
large-dimensional linear regression and change point detection when change
points and additive outliers exist simultaneously. Simulations and empirical
analysis are conducted to illustrate the good performance of our methods.
