Approximations to Bayesian Sequential Tests of Composite Hypotheses

This paper deals with approximations to Bayesian sequential tests of composite hypotheses. If the distributions of the data form an exponential or truncation family, then such tests may be described by a continuation region in the space of $n$, the sample size, and $M_n$, the sufficient statistics, which are of fixed dimension. In this case Schwarz has been able to describe the asymptotic shape of the continuation region as the sampling cost $c$ approaches zero. We have generalized Schwarz's work by considering more general families of distributions. In this paper the role of $M_n$ is played by the log likelihood function, and we show that the optimal Bayesian stopping rule may be approximated by a stopping rule which depends only on $n, c,$ and two likelihood ratio test statistics.