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

## TL;DR

This paper studies robust estimation of a probability distribution's mean on the simplex under adversarial contamination, establishing minimax rates for various distances and providing confidence regions that adapt to sparsity.

## Contribution

It derives minimax rates for outlier-robust mean estimation on the probability simplex across multiple distances and constructs adaptive confidence regions.

## Key findings

- Minimax rates differ for total-variation, Hellinger, and L2 distances.
- Sample average attains all these minimax rates.
- Confidence regions shrink at the minimax rate and adapt to sparsity.

## Abstract

We consider the problem of estimating the mean of a distribution supported by the $k$-dimensional probability simplex in the setting where an $\varepsilon$ fraction of observations are subject to adversarial corruption. A simple particular example is the problem of estimating the distribution of a discrete random variable. Assuming that the discrete variable takes $k$ values, the unknown parameter $\boldsymbol \theta$ 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 $\mathbb L^2$-distance between two probability measures. We also provide confidence regions for the unknown mean that shrink at the minimax rate. Our analysis reveals that the minimax rates associated to these three distances are all different, but they are all attained by the sample average. Furthermore, we show that the latter is adaptive to the possible sparsity of the unknown vector. Some numerical experiments illustrating our theoretical findings are reported.