The minimax optimality of the Maximum Likelihood Estimator (MLE) is a fundamental question in mathematical statistics. For Donsker classes, it is well-known that the MLE is minimax optimal. However, in the non-Donsker regime, the MLE may be minimax sub-optimal. In this talk, we present new techniques to evaluate the statistical performance of the MLE in the non-Donsker regime. As an application, we demonstrate that the log-concave MLE is optimal in all dimensions. In comparison, for convex regression, the MLE can be suboptimal when d ≥ 5.
This talk is based on joint works with Dagan, Gao, Guntuboyina, Rakhlin and Sen.
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