Some topics in sequential analysis

Sequential analysis refers to the statistical theory and methods that can be applied to situations where the sample size is not fixed in advance. Instead, the data are collected sequentially over time, and the sampling is stopped according to a pre-specified stopping rule as soon as the accumulated information is deemed sufficient. The goal of this adaptive approach is to reach a reliable decision as soon as possible. This dissertation investigates two problems in sequential analysis. In the first problem, assuming that data are collected sequentially from independent streams, we consider the simultaneous testing of multiple hypotheses. We start with the class of procedures that control the classical familywise error probabilities of both type I and type II under two general setups: when the number of signals (correct alternatives) is known in advance, and when we only have a lower and an upper bound for it. Then we continue to study two generalized error metrics: under the first one, the probability of at least k mistakes, of any kind, is controlled; under the second, the probabilities of at least k1 false positives and at least k2 false negatives are simultaneously controlled. For each formulation, the optimal expected sample size is characterized, to a first-order asymptotic approximation as the error probabilities vanish, and a novel multiple testing procedure is proposed and shown to be asymptotically efficient under every signal configuration. In the second problem, we propose a generalization of the Bayesian sequential change detection problem, where the change is a latent event that should be not only detected but also accelerated. It is assumed that the sequentially collected observations are responses to treatments selected in real time. The assigned treatments not only determine the distribution of responses before and after the change, but also influence when the change happens. The problem is to find a treatment assignment rule and a stopping rule to minimize the average total number of observations subject to a bound on the false-detection probability. We propose an intuitive solution, which is easy to implement and achieves for a large class of change-point models the optimal performance up to a first-order asymptotic approximation. A simulation study suggests the almost exact optimality of the proposed scheme under a Markovian change-point model