MPS-RR 1998-29

November 1998

# Solution of the Bayesian Sequential Testing Problem for a Poisson Process

by:

### Albert N. Shiryaev

We present the explicit solution of the Bayesian problem of sequential testing
of two simple hypotheses about the intensity of an observed Poisson process.
The method of proof consists of reducing the initial problem to a free-boundary
differential-difference Stephan problem, and solving the latter by use of the principles
of smooth and continous fit. The principle of continous fit emerges as some
unexpected novelty; in this context playing the key role. The rigorous proof of the
optimality of the Wald sequential probability ratio test in the variational formulation
of the same sequential problem is obtained as an immediate consequence.

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This paper has now been published in *Expanded version appears under the title "Sequential testing problems for Poisson processes" in Ann. Statist. 28, 837-859 (2000)*