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

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