Minimax rates in outlier-robust estimation of discrete models

We consider the problem of estimating the probability distribution of a discrete random variable in the setting where the observations are corrupted by outliers. Assuming that the discrete variable takes k values, the unknown parameter p is a k-dimensional vector belonging to the probability simplex. We first describe various settings of contamination and discuss the relation between these settings. We then establish minimax rates when the quality of estimation is measured by the total-variation distance, the Hellinger distance, or the L2-distance between two probability measures. Our analysis reveals that the minimax rates associated to these three distances are all different, but they are all attained by the maximum likelihood estimator. Note that the latter is efficiently computable even when the dimension is large. Some numerical experiments illustrating our theoretical findings are reported.