This thesis investigates variable stage size multistage hypothesis testing in three different contexts, each building on the previous.We first consider the problem of sampling a random process in stages until it crosses a predetermined boundary at the end of a stage -- first for Brownian motion and later for a sum of i.i.d. random variables. A multistage sampling procedure is derived and its properties are shown to be not only sufficient but also necessary for asymptotic optimality as the distance to the boundary goes to infinity.Next we consider multistage testing of two simple hypotheses about the unknown parameter of an exponential family. Tests are derived, based on optimal multistage sampling procedures, and are shown to be asymptotically optimal.Finally we consider multistage testing of two separated composite hypotheses about the unknown parameter of an exponential family. Tests are derived, based on optimal multistage tests of simple hypotheses, and are shown to be asymptotically optimal. Numerical simulations show marked improvement over group sequential sampling in both the simple and composite hypotheses contexts.
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