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In this work we construct an optimal shrinkage estimator for the precisionmatrix in high dimensions. We consider the general asymptotics when the numberof variables $p\rightarrow\infty$ and the sample size $n\rightarrow\infty$ sothat $p/n\rightarrow c\in (0, +\infty)$. The precision matrix is estimateddirectly, without inverting the corresponding estimator for the covariancematrix. The recent results from the random matrix theory allow us to find theasymptotic deterministic equivalents of the optimal shrinkage intensities andestimate them consistently. The resulting distribution-free estimator hasalmost surely the minimum Frobenius loss. Additionally, we prove that theFrobenius norms of the inverse and of the pseudo-inverse sample covariancematrices tend almost surely to deterministic quantities and estimate themconsistently. At the end, a simulation is provided where the suggestedestimator is compared with the estimators for the precision matrix proposed inthe literature. The optimal shrinkage estimator shows significant improvementand robustness even for non-normally distributed data.

Optimal Linear Shrinkage Estimator for Large Dimensional Precision Matrix