  The spiked covariance model has gained increasing popularity in
high-dimensional data analysis. A fundamental problem is determination of the
number of spiked eigenvalues, $K$. For estimation of $K$, most attention has
focused on the use of $top$ eigenvalues of sample covariance matrix, and there
is little investigation into proper ways of utilizing $bulk$ eigenvalues to
estimate $K$. We propose a principled approach to incorporating bulk
eigenvalues in the estimation of $K$. Our method imposes a working model on the
residual covariance matrix, which is assumed to be a diagonal matrix whose
entries are drawn from a gamma distribution. Under this model, the bulk
eigenvalues are asymptotically close to the quantiles of a fixed parametric
distribution. This motivates us to propose a two-step method: the first step
uses bulk eigenvalues to estimate parameters of this distribution, and the
second step leverages these parameters to assist the estimation of $K$. The
resulting estimator $\hat{K}$ aggregates information in a large number of bulk
eigenvalues. We show the consistency of $\hat{K}$ under a standard spiked
covariance model. We also propose a confidence interval estimate for $K$. Our
extensive simulation studies show that the proposed method is robust and
outperforms the existing methods in a range of scenarios. We apply the proposed
method to analysis of a lung cancer microarray data set and the 1000 Genomes
data set.
