  A sample covariance matrix $\boldsymbol{S}$ of completely observed data is
the key statistic in a large variety of multivariate statistical procedures,
such as structured covariance/precision matrix estimation, principal component
analysis, and testing of equality of mean vectors. However, when the data are
partially observed, the sample covariance matrix from the available data is
biased and does not provide valid multivariate procedures. To correct the bias,
a simple adjustment method called inverse probability weighting (IPW) has been
used in previous research, yielding the IPW estimator. The estimator plays the
role of $\boldsymbol{S}$ in the missing data context so that it can be plugged
into off-the-shelf multivariate procedures. However, theoretical properties
(e.g. concentration) of the IPW estimator have been only established under very
simple missing structures; every variable of each sample is independently
subject to missing with equal probability.
  We investigate the deviation of the IPW estimator when observations are
partially observed under general missing dependency. We prove the optimal
convergence rate $O_p(\sqrt{\log p / n})$ of the IPW estimator based on the
element-wise maximum norm. We also derive similar deviation results even when
implicit assumptions (known mean and/or missing probability) are relaxed. The
optimal rate is especially crucial in estimating a precision matrix, because of
the "meta-theorem" that claims the rate of the IPW estimator governs that of
the resulting precision matrix estimator. In the simulation study, we discuss
non-positive semi-definiteness of the IPW estimator and compare the estimator
with imputation methods, which are practically important.
