  Principal component analysis (PCA) is a widely used method for dimension
reduction. In high dimensional data, the "signal" eigenvalues corresponding to
weak principal components (PCs) do not necessarily separate from the bulk of
the "noise" eigenvalues. Therefore, popular tests based on the largest
eigenvalue have little power to detect weak PCs. In the special case of the
spiked model, certain tests asymptotically equivalent to linear spectral
statistics (LSS)---averaging effects over all eigenvalues---were recently shown
to achieve some power.
  We consider a nonparametric, non-Gaussian generalization of the spiked model
to the setting of Marchenko and Pastur (1967). This allows a general bulk of
the noise eigenvalues, accomodating correlated variables even under the null
hypothesis of no significant PCs.
  We develop new tests based on LSS to detect weak PCs in this model. We show
using the CLT for LSS that the optimal LSS satisfy a Fredholm integral equation
of the first kind. We develop algorithms to solve it, building on our recent
method for computing the limit empirical spectrum. In contrast to the standard
spiked model, we find that under "widely spread" null eigenvalue distributions,
the new tests have a lot of power.
