Open Access
2020 Confidence regions and minimax rates in outlier-robust estimation on the probability simplex
Amir-Hossein Bateni, Arnak S. Dalalyan
Electron. J. Statist. 14(2): 2653-2677 (2020). DOI: 10.1214/20-EJS1731


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.


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Amir-Hossein Bateni. Arnak S. Dalalyan. "Confidence regions and minimax rates in outlier-robust estimation on the probability simplex." Electron. J. Statist. 14 (2) 2653 - 2677, 2020.


Received: 1 January 2020; Published: 2020
First available in Project Euclid: 18 July 2020

zbMATH: 1447.62031
MathSciNet: MR4124558
Digital Object Identifier: 10.1214/20-EJS1731

Primary: 62F35
Secondary: 62H12

Keywords: Confidence regions , discrete models , robust estimation

Vol.14 • No. 2 • 2020
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