## Electronic Journal of Statistics

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

#### 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.

#### Article information

Source
Electron. J. Statist., Volume 14, Number 2 (2020), 2653-2677.

Dates
First available in Project Euclid: 18 July 2020

https://projecteuclid.org/euclid.ejs/1595037616

Digital Object Identifier
doi:10.1214/20-EJS1731

Subjects
Primary: 62F35: Robustness and adaptive procedures
Secondary: 62H12: Estimation

#### Citation

Bateni, Amir-Hossein; Dalalyan, Arnak S. Confidence regions and minimax rates in outlier-robust estimation on the probability simplex. Electron. J. Statist. 14 (2020), no. 2, 2653--2677. doi:10.1214/20-EJS1731. https://projecteuclid.org/euclid.ejs/1595037616

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