  Portfolio managers faced with limited sample sizes must use factor models to
estimate the covariance matrix of a high-dimensional returns vector. For the
simplest one-factor market model, success rests on the quality of the estimated
leading eigenvector "beta".
  When only the returns themselves are observed, the practitioner has available
the "PCA" estimate equal to the leading eigenvector of the sample covariance
matrix. This estimator performs poorly in various ways. To address this problem
in the high-dimension, limited sample size asymptotic regime and in the context
of estimating the minimum variance portfolio, Goldberg, Papanicolau, and
Shkolnik developed a shrinkage method (the "GPS estimator") that improves the
PCA estimator of beta by shrinking it toward a constant target unit vector.
  In this paper we continue their work to develop a more general framework of
shrinkage targets that allows the practitioner to make use of further
information to improve the estimator. Examples include sector separation of
stock betas, and recent information from prior estimates. We prove some precise
statements and illustrate the resulting improvements over the GPS estimator
with some numerical experiments.
