Translator Disclaimer
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

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.

Citation

Download Citation

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. https://doi.org/10.1214/20-EJS1731

Information

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

Subjects:
Primary: 62F35
Secondary: 62H12

JOURNAL ARTICLE
25 PAGES


SHARE
Vol.14 • No. 2 • 2020
Back to Top