Electronic Journal of Statistics

A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation

Ramji Venkataramanan and Oliver Johnson

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In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to do this involves the use of Fano’s inequality. However, recent work in an information-theoretic setting has shown that an argument based on binary hypothesis testing gives tighter converse results (error lower bounds) than Fano for channel coding problems. We adapt this technique to the statistical setting, and argue that Fano’s inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case.

Article information

Electron. J. Statist., Volume 12, Number 1 (2018), 1126-1149.

Received: June 2017
First available in Project Euclid: 27 March 2018

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62B10: Information-theoretic topics [See also 94A17] 62G07: Density estimation

Minimax lower bounds Fano’s inequality compressed sensing density estimation active learning

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Venkataramanan, Ramji; Johnson, Oliver. A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation. Electron. J. Statist. 12 (2018), no. 1, 1126--1149. doi:10.1214/18-EJS1419. https://projecteuclid.org/euclid.ejs/1522116041

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