Estimating High-dimensional Covariance and Precision Matrices under General Missing Dependence

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.