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Biometrika 1966 53(3-4):455-467; doi:10.1093/biomet/53.3-4.455
© 1966 by Biometrika Trust
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The outer needle of some Bayes sequential continuation regions

JOHN W. PRATT

Harvard University

This paper explores analytically the outer portion of the Bayes continuation region in sequential sampling problems for the Bernoulli process with a beta distribution on the parameter p, and for the Poisson process with a scaled gamma distribution on the parameter {lambda}. We assume that there are two actions, that the cost per Bernoulli trial or per unit time of Poisson observation is constant, and that utility is suitably linear. Our concern is with continuation points which cannot be reached from any continuation point corresponding to a smaller number of Bernoulli successes or Poisson occurrences. These points form an inaccessible region which need not be considered in calculating the rest of the continuation region. In length this region is approximately 50–63% of the whole continuation region when the sampling cost is small, 63% in the Poisson case. We also give computable nearly tight bounds on the inner end of the region, which may be used to start backward induction to obtain the rest of the continuation region. These results follow from more detailed results concerning the outer portion of continuation region.


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