  This paper aims to test the number of spikes in a generalized spiked
covariance matrix, the spiked eigenvalues of which may be extremely larger or
smaller than the non-spiked ones. For a high-dimensional problem, we first
propose a general test statistic and derive its central limit theorem by random
matrix theory without a Gaussian population constraint. We then apply the
result to estimate the noise variance and test the equality of the smallest
roots in generalized spiked models. Simulation studies showed that the proposed
test method was correctly sized, and the power outcomes showed the robustness
of our statistic to deviations from a Gaussian population. Moreover, our
estimator of the noise variance resulted in much smaller mean absolute errors
and mean squared errors than existing methods. In contrast to previously
developed methods, we eliminated the strict conditions of diagonal or
block-wise diagonal form of the population covariance matrix and extend the
work to a wider range without the assumption of normality. Thus, the proposed
method is more suitable for real problems.
