The Annals of Statistics

Complete Classes for Sequential Tests of Hypotheses

L. D. Brown, Arthur Cohen, and W. E. Strawderman

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Abstract

We consider problems of sequential testing when the loss function is the sum of a component due to an error in the terminal decision and a cost of observation component. In all cases we establish a characterization of a complete class or an essentially complete class. In order to obtain such results for testing a null hypothesis against an alternative hypothesis we establish complete class results for testing the closure of the null hypothesis against the closure of the alternative hypothesis. A complete class for testing closure of null against closure of alternative is an essentially complete class for testing null against alternative. Furthermore, a complete class for testing closure of null against closure of alternative is a complete class for testing null against alternative when the risks have certain continuity properties. Such continuity properties do hold in many cases. Three models are treated. The first is when the closure of the null space is compact and the cost of the first observation is positive. Under very unrestrictive conditions it is shown that the Bayes tests form a complete class. This result differs considerably from most fixed sample analogues that have been studied. The second model is when the closure of the null space is compact, the distributions are exponential family, and the cost of the first observation is zero. The third model is for the one dimensional exponential family case when the hypotheses are one sided.

Article information

Source
Ann. Statist., Volume 8, Number 2 (1980), 377-398.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344959

Digital Object Identifier
doi:10.1214/aos/1176344959

Mathematical Reviews number (MathSciNet)
MR560735

Zentralblatt MATH identifier
0429.62058

JSTOR
links.jstor.org

Subjects
Primary: 62L10: Sequential analysis
Secondary: 62C07: Complete class results

Keywords
Sequential tests complete classes Bayes tests generalized Bayes tests admissibility stopping rules

Citation

Brown, L. D.; Cohen, Arthur; Strawderman, W. E. Complete Classes for Sequential Tests of Hypotheses. Ann. Statist. 8 (1980), no. 2, 377--398. doi:10.1214/aos/1176344959. https://projecteuclid.org/euclid.aos/1176344959


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Corrections

  • See Correction: L. D. Brown, Arthur Cohen, W. E. Strawderman. Correction: Complete Classes for Sequential Tests of Hypotheses. Ann. Statist., Volume 17, Number 3 (1989), 1414--1416.