  Missing data occur frequently in a wide range of applications. In this paper,
we consider estimation of high-dimensional covariance matrices in the presence
of missing observations under a general missing completely at random model in
the sense that the missingness is not dependent on the values of the data.
Based on incomplete data, estimators for bandable and sparse covariance
matrices are proposed and their theoretical and numerical properties are
investigated.
  Minimax rates of convergence are established under the spectral norm loss and
the proposed estimators are shown to be rate-optimal under mild regularity
conditions. Simulation studies demonstrate that the estimators perform well
numerically. The methods are also illustrated through an application to data
from four ovarian cancer studies. The key technical tools developed in this
paper are of independent interest and potentially useful for a range of related
problems in high-dimensional statistical inference with missing data.
