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We consider the problem of estimating the mean of a distribution supported bythe $k$-dimensional probability simplex in the setting where an $\varepsilon$fraction of observations are subject to adversarial corruption. A simpleparticular example is the problem of estimating the distribution of a discreterandom variable. Assuming that the discrete variable takes $k$ values, theunknown parameter $\boldsymbol \theta$ is a $k$-dimensional vector belonging tothe probability simplex. We first describe various settings of contaminationand discuss the relation between these settings. We then establish minimaxrates when the quality of estimation is measured by the total-variationdistance, the Hellinger distance, or the $\mathbb L^2$-distance between twoprobability measures. We also provide confidence regions for the unknown meanthat shrink at the minimax rate. Our analysis reveals that the minimax ratesassociated to these three distances are all different, but they are allattained by the sample average. Furthermore, we show that the latter isadaptive to the possible sparsity of the unknown vector. Some numericalexperiments illustrating our theoretical findings are reported.

Confidence regions and minimax rates in outlier-robust estimation on the probability simplex