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

Ramji Venkataramanan, Oliver Johnson

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

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.
MSC 2010 subject classifications: 62G05, 62B10, 62G07.
Keywords and phrases: minimax lower bounds, Fano’s inequality, compressed
sensing, density estimation, active learning.
Original languageEnglish
Pages (from-to)1126-1149
Number of pages24
JournalElectronic Journal of Statistics
Volume12
Issue number1
Early online date27 Mar 2018
DOIs
Publication statusPublished - 2018

Keywords

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

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