Asymptotic approximations for error probabilities of sequential or fixed sample size tests in exponential families

Asymptotic approximations for the error probabilities of sequential tests o f composite hypotheses in multiparameter exponential families are developed herein for a general class of test statistics, including generalized likel ihood ratio statistics and other functions of the sufficient statistics. Th ese results not only generalize previous approximations for Type I error pr obabilities of sequential generalized likelihood ratio tests, but also prov ide a unified treatment of both sequential and fixed sample size tests and of Type I and Type II error probabilities. Geometric arguments involving in tegration over tubes play an important role in this unified theory.