  In this paper we study covariance estimation with missing data. We consider
missing data mechanisms that can be independent of the data, or have a time
varying dependency. Additionally, observed variables may have arbitrary (non
uniform) and dependent observation probabilities. For each mechanism, we
construct an unbiased estimator and obtain bounds for the expected value of
their estimation error in operator norm. Our bounds are equivalent, up to
constant and logarithmic factors, to state of the art bounds for complete and
uniform missing observations. Furthermore, for the more general non uniform and
dependent cases, the proposed bounds are new or improve upon previous results.
Our error estimates depend on quantities we call scaled effective rank, which
generalize the effective rank to account for missing observations. All the
estimators studied in this work have the same asymptotic convergence rate (up
to logarithmic factors).
