Electronic Journal of Statistics

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

Amir-Hossein Bateni and Arnak S. Dalalyan

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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

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

Received: January 2020
First available in Project Euclid: 18 July 2020

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Digital Object Identifier

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

Robust estimation discrete models confidence regions

Creative Commons Attribution 4.0 International License.


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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