The Annals of Statistics

Recursive Testing of Multiple Hypotheses: Consistency and Efficiency of the Bayes Rule

Andrew L. Rukhin

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A version of the multiple hypotheses testing problem is studied in which the decision procedure is based only on the current observation and the previous decision. Conditions for inconsistency and consistency of the stepwise Bayes rule, which are related to the boundedness of the likelihood ratios, are given. The (typically slow) rate of convergence of the error probabilities of consistent procedures is determined, and a sharp lower bound for the Bayes risk in terms of bounds on the likelihood ratios is derived. A modification of the recursive Sakrison's procedure for a continuous estimation problem is obtained in this setting by embedding the discrete family of original probability distributions into an exponential family.

Article information

Ann. Statist., Volume 22, Number 2 (1994), 616-633.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F05: Asymptotic properties of tests
Secondary: 62F12: Asymptotic properties of estimators 62E20: Asymptotic distribution theory

Asymptotic efficiency consistency error probabilities exponential family likelihood ratios multiple hypothesis testing recursive Bayes rule Sakrison's estimator


Rukhin, Andrew L. Recursive Testing of Multiple Hypotheses: Consistency and Efficiency of the Bayes Rule. Ann. Statist. 22 (1994), no. 2, 616--633. doi:10.1214/aos/1176325487.

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