Optimal Uncertainty Size in Distributionally Robust Inverse Covariance Estimation
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In a recent paper, Nguyen, Kuhn, and Esfahani (2018) built a distributionally robust estimator for the precision matrix (i.e. inverse covariance matrix) of a multivariate Gaussian distribution. The distributional uncertainty size is a key ingredient in the construction of such estimator, which is shown to have an excellent empirical performance. In this paper, we develop a statistical theory which shows how to optimally choose the uncertainty size to minimize the associated Stein loss. Surprisingly, the optimal uncertainty size scales linearly with the sample size, instead of the canonical square-root scaling which may be expected for this problem.
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