  Estimating the eigenvalues of a population covariance matrix from a sample
covariance matrix is a problem of fundamental importance in multivariate
statistics; the eigenvalues of covariance matrices play a key role in many
widely techniques, in particular in Principal Component Analysis (PCA). In many
modern data analysis problems, statisticians are faced with large datasets
where the sample size, n, is of the same order of magnitude as the number of
variables p. Random matrix theory predicts that in this context, the
eigenvalues of the sample covariance matrix are not good estimators of the
eigenvalues of the population covariance.
  We propose to use a fundamental result in random matrix theory, the
Marcenko-Pastur equation, to better estimate the eigenvalues of large
dimensional covariance matrices. The Marcenko-Pastur equation holds in very
wide generality and under weak assumptions. The estimator we obtain can be
thought of as "shrinking" in a non linear fashion the eigenvalues of the sample
covariance matrix to estimate the population eigenvalue. Inspired by ideas of
random matrix theory, we also suggest a change of point of view when thinking
about estimation of high-dimensional vectors: we do not try to estimate
directly the vectors but rather a probability measure that describes them. We
think this is a theoretically more fruitful way to think statistically about
these problems.
  Our estimator gives fast and good or very good results in extended
simulations. Our algorithmic approach is based on convex optimization. We also
show that the proposed estimator is consistent.
